stochastic oscillations in living cells

Stochastic Oscillations in Living Cells

Exemplified by Ca2+ and p53

D I S S E R T A T I O N

zur Erlangung des akademischen Grades

d o c t o r r e r u m n a t u r a l i u m(Dr. rer. nat.)

im Fach Biophysik

eingereicht an derLebenswissenschaftlichen Fakultät

Humboldt-Universität zu Berlin

vonDipl. Physiker Gregor Mönke

Präsident der Humboldt-Universität zu Berlin:Prof. Dr. Jan-Hendrik Olbertz

Dekan der Lebenswissenschaftlichen Fakultät:Prof. Dr. Richard Lucius

Gutachter:1. Prof. Hanspeter Herzel2. Dr. Alexander Löwer3. Dr. Martin Falcke

eingereicht am: 18. Dezember 2014Tag der mündlichen Prüfung: 7. Mai 2015

Abstract

Recent developments in cell biology, especially single cell experiments,allow for a very detailed view on intracellular processes. It has becomeevident, that there are many complex dynamics present, which can not bedescribed by standard steady state or regular oscillatory regimes. In thiswork two signaling pathways, involving the tumor suppressor p53 and thesecond messenger Ca2+ , are discussed and modelled in this context.

The tumor suppressor p53 shows a pulsatile response in single cellsafter induction of DNA double strand breaks (DSBs). Except for veryhigh amounts of damage, these pulses appear at irregular times. Toquantify these irregularities, the inter pulse interval (IPI) distributionsare extracted from the data with the help of a wavelet based featuredetection. A comparison to synthetic noisy limit cycle oscillator dynamicsindicate the presence of non-oscillatory regimes in the data. The concept ofexcitable systems is employed as a convenient way to model such observeddynamics. An application to biomolecular reaction networks shows the needfor a positive feedback within the p53 regulatory network. Exploiting thereported ultrasensitive dynamics of the upstream damage sensor kinases,leads to a simplified excitable kinase-phosphatase model. Coupling that tothe canonical negative feedback p53 regulatory loop, is the core idea behindthe construction of the excitable p53 model. A detailed bifurcation analysisof the full model establishes a robust excitable regime, which can be switchedto oscillatory dynamics via a strong DNA damage signal. This elegantlyreproduces the experimentally observed digital response after damage.To further assess the observed cell-to-cell variability, single cell damagefoci trajectories are analysed. A Markovian process describing the DSBdynamics is constructed and a strategy to extract its parameters from thedata is devised. Driving the p53 model with that stochastic process yieldspulsatile dynamics which reflect different experimental scenarios. Finally,the modeling results lead to a reanalysis of published data, supporting theanticipated structure of the regulatory p53 network.

Intracellular Ca2+ concentration spikes arise from a hierarchic cascade ofstochastic events. In this thesis, the emphasis first lies on the mathematicaltheory behind the established stochastic Ca2+ models. An analyticalsolution strategy, employing a semi-Markovian description and involvingLaplace transformations, is devised and successfully applied to a specificCa2+ model. The new gained insights are then used, to construct a newgeneric Ca2+ model, which elegantly captures many known features of Ca2+

signaling. In particular the experimentally observed relations between theaverage and the standard deviation of the inter spike intervals (ISIs) canbe explained in a concise way. Additionally, an exact and fast algorithm tosimulate semi-Markovian processes is developed. Finally, the theoreticalconsiderations allow to calculate the stimulus encoding relation, whichgoverns the adaption of the Ca2+ signals to varying extracellular stimuli.This is predicted to be a fold change response and new experimental resultsobtained by collaborators display a strong support of this idea.

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Zusammenfassung

Neueste Entwicklungen der Zellbiologie, insbesondere Einzelzell basierteExperimente, erlauben einen sehr genauen Blick auf intrazelluläre Prozesse.Es hat sich gezeigt, dass es sehr vielfältige und komplexe Dynamiken gibt,welche nicht durch stationäre Zustände oder regelmäßige Oszillationenmodelliert werden können. In dieser Arbeit werden zwei intrazelluläreSignalwege, betreffend den Tumorsuppressor p53 und das SignalmolekülCa2+ , in dieser Hinsicht diskutiert und modelliert.

Einzelzellmessungen des Tumorsuppressors p53 zeigen pulsatile Antwor-ten nach Zufügung von DNA Doppelstrangbrüchen (DSBs). Außer für sehrhohe Schadensdosen, ist das zeitliche auftreten dieser Pulse unregelmäßig.Mithilfe eines Wavelet basierten Pulsdetektors werden die einzelzell Trajek-torien untersucht und die inter-Puls Intervall (IPI) Verteilungen extrahiert.Diese weisen auf nicht-oszillatorische Regime in den Daten hin, quantita-tiv überprüft durch den Vergleich mit künstlich erzeugten verrauschtenOszillationen. Die Theorie der anregbaren Systeme angewendet auf regu-latorische Netzwerke ermöglicht dieses komplexe Verhalten mathematischzu beschreiben. Die Theorie erfordert zwingend einen starken positivenfeedback im modellierten p53 Netzwerk. Das beobachtete ultra-sensitiveVerhalten der Schadens-Sensor Kinasen, deutet auf einen solchen positivenfeedback hin. Die Kopplung dieser Kinase Dynamik mit dem kanonischenp53 negativen feedback loop, ergibt ein anregbares p53 Modell. Detaillier-te Bifurkationsanalysen zeigen ein robustes anregbares Regime, welchesdurch ein starkes Schadenssignal auch in Oszillationen überführt werdenkann. Dies zusammen reproduziert auf natürliche Weise die experimentellbeobachtete digitale Antwort nach Schaden. Desweiteren werden einzel-zell Schadensdynamiken analysiert. Diese zeigen eine große Variabilitätzwischen den Zellen, was zu einer markovschen Beschreibung der DSBDynamik führt. Sowohl die Reparatur- als auch die spontane Bruchratekönnen aus den Daten geschätzt werden. Treibt man das p53 Modell mitdiesem stochastischen Prozess, kann sowohl das oszillatorische Verhaltennach hohem Schaden, als auch das unregelmäßige pulsatile Verhalten ohneäußere Stimulation reproduziert werden. Schließlich führen diese Überle-gungen zu einer Reanalyse bereits publizierter Daten, welche die gemachtenAnnahmen über das regulatorische p53 Netzwerk unterstützen.

Intrazelluläre Ca2+ Spikes entstehen durch eine hierarchische Kaskadestochastischer prozesse. In dieser Arbeit wird zuerst die hinter etabliertenstochastischen Ca2+ Modellen liegende mathematische Theorie vertieft.Die Anwendung einer semi-markovschen Beschreibung führt zu prakti-schen analytischen Lösungen mit Hilfe von Laplace Transformationen. Einehierbei entdeckte Zeitskalenseparation ermöglicht ein neues allgemeinesCa2+ -Modell. Dieses erklärt auf äußerst prägnante Weise viele wesentlicheexperimentelle Ergebnisse, insbesondere die Momentenbeziehungen derinter-Spike Intervall Verteilungen. Numerische Analysen bestätigen dieValidität dieses neuen Modells. Schließlich erlaubt die hier vorgestellteTheorie Berechnungen der Stimulus-Enkodierung, also die Adaption desCa2+ Signals auf veränderliche extrazelluläre Stimuli. Die Vorhersage ei-ner fold change Enkodierung kann durch Experimente, durchgeführt vonKollaborationspartnern, gestützt werden.

v

Dann lächelt der bescheidene Gelehrte und sagt zu den Staunenden:“Was habe ich denn erschaffen? Nichts. Der Mensch erfindet keine Kraft, er lenktsie nur, und die Wissenschaft besteht darin, die Natur nachzuahmen.”

Honoré de Balzac

vii

Contents

1. An excitable p53 model 11.1. Introduction to p53 and the DNA damage response system . . . 1

1.1.1. The guardian of the genome . . . . . . . . . . . . . . . . . 11.1.2. DNA double strand breaks . . . . . . . . . . . . . . . . . 21.1.3. Established p53 dynamics . . . . . . . . . . . . . . . . . . 2

1.2. Experimental findings not covered by oscillatory dynamics . . . . 41.2.1. Introducing IPI distributions . . . . . . . . . . . . . . . . 51.2.2. p53 data analysis . . . . . . . . . . . . . . . . . . . . . . . 7

1.3. Modeling theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1. Preface - dynamical systems theory and biochemical re-

action networks . . . . . . . . . . . . . . . . . . . . . . . . 121.3.2. Negative feedback oscillators . . . . . . . . . . . . . . . . 141.3.3. Introduction to excitability - case study FitzHugh Nagumo 18

1.4. The ATM-Wip1 switch building block . . . . . . . . . . . . . . . 251.4.1. ATM as Signalling Switch . . . . . . . . . . . . . . . . . . 251.4.2. Incorporating the Phosphatase Wip1 . . . . . . . . . . . . 27

1.5. Including the core negative feedback loop - the full p53 model . . 301.5.1. The effectual modeled p53 network for the DSB response 301.5.2. Bifurcation analysis and deterministic dynamics . . . . . 33

1.6. Driving the p53 model with a stochastic DSB process . . . . . . 391.6.1. Constructing a stochastic process for the DSB dynamics . 391.6.2. Stochastic forcing of the excitable p53 model . . . . . . . 46

1.7. Reanalysis of inhibitor experiments . . . . . . . . . . . . . . . . . 501.8. Discussing the p53 modeling approach . . . . . . . . . . . . . . . 54

2. Hierarchic stochastic modelling of intracellular Ca2+ 592.1. Introduction to intracellular Ca2+ signaling . . . . . . . . . . . . 592.2. An analytical approach to hierarchic stochastic modelling . . . . 61

2.2.1. What is HSM ? . . . . . . . . . . . . . . . . . . . . . . . . 612.2.2. Semi-Markov processes . . . . . . . . . . . . . . . . . . . . 632.2.3. Non-Markovian master equations and first passage times . 642.2.4. A simple semi-Markovian system . . . . . . . . . . . . . . 652.2.5. Explicit solutions for the tetrahedron Ca2+ model . . . . 69

2.3. Exploiting time scale separation - The generic Ca2+ model . . . . 742.3.1. Model Construction . . . . . . . . . . . . . . . . . . . . . 752.3.2. Results of the generic model . . . . . . . . . . . . . . . . . 772.3.3. Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.4. Numerical Analysis of the HSM Ca2+ model . . . . . . . . . . . . 812.4.1. The DSSA algorithm . . . . . . . . . . . . . . . . . . . . . 81

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Contents

2.4.2. An exact semi-Markovian simulation algorithm . . . . . . 832.5. Encoding Stimulus intensities in random spike trains . . . . . . . 86

2.5.1. Theoretical predictions . . . . . . . . . . . . . . . . . . . . 862.5.2. Experiments supporting the fold change encoding hypothesis 89

2.6. Discussion of the stochastic Ca2+ modeling . . . . . . . . . . . . 93

3. Concluding remarks 97

A. Appendix p53 99A.1. Peak detection with wavelets . . . . . . . . . . . . . . . . . . . . 99A.2. Table of Parameters for the p53 model . . . . . . . . . . . . . . . 105A.3. Sensitivity of pulse shapes and the excitation threshold on pa-

rameter variations . . . . . . . . . . . . . . . . . . . . . . . . . . 107A.4. Codimension-2 bifurcation diagrams . . . . . . . . . . . . . . . . 112A.5. Period of oscillations in the p53 model . . . . . . . . . . . . . . . 113

B. Appendix Ca2+ 115B.1. Laplace transformations of Ψo and Ψc . . . . . . . . . . . . . . . 115

C. List of abbreviations 119

Acknowledgments 131

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1. An excitable p53 model

1.1. Introduction to p53 and the DNA damage responsesystem

The mammalian p53 gene belongs, with tens of thousands of publications, tothe best studied genes in molecular biology. It is inactivated in about half of thehuman cancers and consequently was the first tumor suppressor to be identified[3]. This insight put p53 right in the center of molecular cancer research, andtriggered an avalanche of research until today [108]. Although, there have beennumerous modeling approaches in the past, the focus on the irregular pulsatiledynamics found for the basal p53 dynamics shed new light on the underlyingdesign principles of the regulatory network for p53. The findings to be developedin the following were greatly supported by the group of Alexander Loewer at theMDC Berlin. All experimental raw data used here was either directly measuredby the group, or originates from publications developed at the group of GalitLahav at the Harvard Medical School with a major contribution from AlexanderLoewer.

1.1.1. The guardian of the genomeAs a central hub in different stress response signalling networks, p53 can beactivated by various upstream kinases which often serve as stress sensors. Itgets activated e.g. by oncogene induced p14arf , in response to single strandDNA breaks and double strand DNA breaks (DSBs). The latter is mediatedby the ataxia telangiectasia mutated (ATM) kinase, which serves as a damagesensor. Upon activation p53 acts as a transcription factor for numerous targetgenes which in effect regulate different cellular stress responses like DNA repair,cell-cycle-arrest or apoptosis [80, 108]. The versatility of the downstream outputof p53 activation stems from its many post translational modification sites.These include, besides phosphorylation, acetylation, sumolation, glycolysationand ubiquitination on various residues [55]. In summary, the tumor suppressingfunction of p53 is achieved by preventing the proliferation of cells with corruptedgenomic integrity. It was therefore called the guardian of the genome [28, 58].

A key feature of the p53 regulatory system is that p53 transcriptionally acti-vates its own suppressor Mdm2 (mouse double minute protein 2) [45, 77]. Mdm2is an E3-ligase which binds to p53 and polyubiquitinates it. This effectivelyflags p53 for the proteolytic pathway and therefore induces its degradation [19].In unstressed conditions this negative feedback loop keeps p53 at low levels asneeded for cell homeostasis. Remarkably the entire regulation of p53 takes placeon a post translational level, as the level of p53 transcripts remains constantover time, particularly also after stimulation [52].

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1.1.2. DNA double strand breaks

DNA double strand breaks are a particularly dangerous form of DNA damage andare a prevalent cause of p53 signaling. If not repaired genomic rearrangementsincluding translocations, deletions and fusion may follow. These kinds of genomicaberrations are often found in cancerous cells [53]. DSBs occur naturally insidecells, most prominent causes are radical metabolic by-products, cosmic radiationand errors during DNA replication. To induce DSBs experimentally living cellsare either irradiated with γ-irradiation or the radiomimetic drug neocarzinostatin(NCS) is used. After damage induction, very complex processes involving damagesensing, repair mechanisms and signalling to downstream pathways includingp53 take place. The protein complexes which develop around a damage locusis called a foci. Some of these processes important for p53 dynamics will bediscussed in detail later in this work in section 1.4.1.

To track foci formation and their subsequent disappearance indicating repairin living cells, a protein called 53BP1 was labeled with the red fluorescent proteinmCherry [63]. 53BP1 is an important mediator of the damage response whichlocalizes early at the damage loci [109]. By using time-lapse microscopy, focitrajectories can be recorded which serve as a proxy for the number of DSBsinside a cell. Foci dynamics for cells stimulated by NCS and γ-irradiation withdifferent dosing are shown in figure 1.2. The repair dynamics generally followan exponential decay, although at least two different molecular mechanismscontribute to the repair process [63]. Highlighted by the quantiles, the variabilityof the amount of initial damage a single cell receives in a fixed dose experimentis quite high. This contributes to the variability in p53 dynamics to be discussedlater.

1.1.3. Established p53 dynamics

The first experimental study which attempted to reveal the p53 dynamics afterDSB induction, in contrast to only steady state transitions observed before,was conducted almost 15 years ago [5]. By harvesting cells every hour afterstimulation and probing for P53 and Mdm2 in a Western Blot analysis theyfound damped oscillatory behavior of both proteins. Interestingly these authorswere also motivated from the theoretical side and proposed an ODE modelcapturing the observed dynamics which was solely based on the properties ofthe P53-Mdm2 negative feedback loop. A detailed discussion of oscillations insystem with only negative feedbacks follows in section 1.3.2.

Since the onset of single cell analysis, the dynamical behavior of cells can bestudied in much more detail. The first reporter system for both P53 and Mdm2was established around 10 years ago [57]. Lahav and her co-workers stablytransfected MCF7 cells with the fusion proteins P53-CFP and Mdm2-YFP.They showed that both fusion constructs were functional and were expressedand therefore regulated like their endogenous counterparts. In summary theywere able to reliably track the P53 and Mdm2 protein dynamics on a singlecell level using time lapse fluorescence microscopy. The main results wereas follows: Mean pulse height and duration are independent of the damage

2

1.1. Introduction to p53 and the DNA damage response system

Figure 1.1.: A fluorescence image of mCherry labeled 53BP1 localized in fociinside the nuclei of four MCF7 cells. DSBs were induced by treatingthe cells with 50ng of the radiomimetic drug NCS.

dose and cells respond with a variable discrete number of such pulses. Thisnumber is dependent on the damage dose, and the authors concluded that thestimulus response is encoded in a digital fashion. Two representative singlecell trajectories after strong stimulation are shown in figure 1.3. Differenttiming and especially different numbers of pulses in individual cells lead todamped oscillations on the population level, as to be seen in a the Westerblot analysis. Subsequently this damped oscillatory behavior as described inthe last paragraph ([5]) could be recovered by averaging over the single celltrajectories. This striking study clearly showed, that single cell analysis canreveal qualitatively different cellular behavior compared to what can be learnedfrom population data. In a subsequent study it was shown, that MCF7 cellscan oscillate for up to three days on a single cell level after DNA damage

3


0 2 4 6 8 10 12

Time (h)

−10

0

10

20

30

40

50N

r.of

foci

50ng NCS25ng NCS

(a) Stimulation by NCS, 179 (25ngNCS) and 220 (50ng NCS) cellsrespectively

0 5 10 15 20 25

Time (h)

−20

0

20

40

60

80

100

Nr.

offo

ci

10 Gy5 Gy

(b) Stimulation by γ-irradiation,63 (10 gray) and 54 (5 gray)cells respectively

Figure 1.2.: Foci dynamics after damage induction. Median number of foci andquantiles in shaded area are shown over time. Raw data obtainedfrom the Loewer lab.

[34]. However, the authors also pointed out, that a significant fraction of thecells showed dynamics unresembling sustained oscillations. Most notably withhigher initial damage dose these irregular trajectories become less abundant. Aquantitative analysis on the irregularity of p53 trajectories measured in weaklyor non-stimulated MCF7 cells is presented in the next section 1.2.2.

1.2. Experimental findings not covered by oscillatorydynamics

The main focus of many works dedicated to study p53 dynamics lays on thesystems response to a high damage induction. Therefore, also many theoreticalstudies concentrate on modeling this high damage scenario. As the mainexperimental results indicate constant amplitude and constant pulse duration, thep53 dynamics after strong stimulation are characterized as sustained oscillations.From the dynamical system theory side, this behavior qualifies to be modeled bylimit cycle oscillators. And indeed most if not all theoretical studies constructan ODE model which exhibits a limit cycle regime [6, 15, 34, 35, 65].

In this section experimental results are presented, which are not readilycaptured by limit cycle models. Most of the analyzed raw data originates fromexperiments done by Alexander Loewer [62] in the group of Galit Lahav. Atfirst, the inter-pulse-interval (IPI) distribution will be introduced, as means toreliably detect p53 dynamics deviating from sustained oscillations. Additionallya generic limit cycle model with additive noise is used to generate syntheticdata to illustrate the IPI distribution characteristics expected for a sustainedoscillator.

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1.2. Experimental findings not covered by oscillatory dynamics

0 5 10 15 20 25

Time (h)

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0 5 10 15 20 25

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ores

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Figure 1.3.: Two representative MCF7 single cell p53 trajectories after strongstimulation with 400ng NCS. The oscillatory dynamics are clearlyvisible. Raw data obtained from the Loewer lab.

1.2.1. Introducing IPI distributions

From a signaling theory viewpoint, the classification of trajectories as showingoscillations involves standard methods like spectral analysis. However, in the realworld of fluorescence single cell p53 data the trajectories are rather short, showingtypically less than ten pulses. In combination with inevitable measurement noise,see e.g. figure 1.3, spectral methods are of limited applicability. Nevertheless, aquantitative measure of oscillatory behavior is needed. The distribution of thecombined IPIs for all cells yields such a measure. To extract the IPIs from thedata, a reliable peak detection algorithm was devised and implemented usingwavelet analysis. Details about it can be found in the Appendix A.1.

The interpretation of the IPI distribution is straightforward. For a perfectoscillator the IPIs are delta distributed δ(t − Tosc), with Tosc being the period ofthe oscillator. By adding a moderate amount of noise to the oscillator resemblingthe variability found in the p53 data, the IPIs are narrowly distributed aroundTosc. To generate synthetic data to illustrate these properties a generic limitcycle oscillator (LCO) [111] given by the following equations:

dr

dt= −λ(r − A) + ξr

dφ

dt=

2π

Tosc+ ξφ,

(1.1)

is used. Here A is the amplitude and λ is the relaxation rate. The two noiseterms ξr and ξφ have Gaussian white noise properties, namely 〈ξi(t + τ), ξi(t)〉 =2Diδ(τ), with i being either the radial variable r or the angular variable φ.The constants Dr and Dφ give the noise intensities for radial and angularperturbations respectively. This stochastic ODE is numerically solved using thestandard Euler-Maruyama method. The generated trajectories are then analyzedby the wavelet based peak detection algorithm, and the IPIs are extracted.

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(a) Section of a representativenoisy LCO trajectory, parame-ters are A = 10, Tosc = 5, λ =0.2, Dr = 1.2 and Dφ = 0.4.

(b) IPI distributions for the noisyLCO for different noise intensi-ties as indicated in the legend.

Figure 1.4.: Dynamics and IPI distributions of the noisy generic LCO introducedin the main text.

In figure 1.4 a representative simulated trajectory of this noisy generic LCOis shown. The corresponding IPI distribution, calculated from a much longersimulation run is shown next to it, together with results for different noiseintensities. As one might intuitively expect, the IPI distributions get broaderwith higher noise intensities. Notably the IPI distributions symmetricallycenter around the LCO period Tosc, which is also clearly visible in the box plotrepresentation shown in figure 1.5.

Figure 1.5.: Box plots of the IPI distributions for the noisy LCO. The distribu-tions are symmetrically centered around Tosc = 5 and their interquartile range decreases with decreasing noise intensity.

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It will become clear in the next section, that it is foremost the uniform pulseshape combined with the aberrant IPI distribution which indicates a non LCObehavior of the p53 dynamics for no or weak stimulation.

1.2.2. p53 data analysisThe raw data reanalyzed here was already published some years ago [7]. Dam-aging the MCF7 cells is done by adding various concentrations of NCS prior tofluorescence microscopy analysis. The cells are then imaged every 20 minuteswhich yields time series of p53 fluorescence intensity. Details of the experimentalconditions are shown in table 1.1. The established results obtained in thepublished works about pulse widths and amplitudes will be re-examined, andthe IPI distributions are evaluated. All analysis is done with the wavelet basedpeak detection method as described in the appendix A.1.

Table 1.1.: Overview of the analyzed p53 data set. The total measurement timeis 48 hours and the sampling rate is 20 minutes.

condition number of cellsControl 92

25ng NCS 10950ng NCS 108100ng NCS 101200ng NCS 88400ng NCS 94

To get a first general overview about how a cell population responds to variousstimulus strengths, pulse counting statistics are shown in figure 1.6. The generalpulsing activity of cells rises with stronger stimulation, which is evident becausehigher stimulation induces on average more DSBs and therefore longer repairtimes. That in term causes p53 activating damage signals to trigger more pulses.

It is, however, noteworthy that there is also a basal p53 dynamic. Even in thecontrol condition there is on average one pulse every twelve hours. One causeof that basal activity are DSBs inflicted during normal cell growth [62], otherdamage sources are spontaneous transient DSBs caused for example by radicalmetabolites. These erratic DSB occurrences will play an important role lateron in this work when a stochastic process describing the DSB dynamics willbe developed. Furthermore, the cell-to-cell variability in the number of pulsesshown in figure 1.7 is very large for cells in identical conditions. This can beat least partly explained by the fact that even for a fixed damage dose there isalways a distribution of actually inflicted DSBs as depicted by the quantiles infigure 1.2 in the preceding section.

Pulse counting alone can not reveal sufficient information about the p53dynamics over time. To further characterize the observed pulsatile dynamics,the IPI distributions are extracted from the data. The IPIs are naturallydependent on the pulse widths, defined here as time between the start of therise and the end of the descent of a pulse as depicted in figure 1.8.

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1 2 3 4 5 6 7 8 9 10 11

Number of pulses

0.00

0.05

0.10

0.15

0.20

0.25

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Fra

ctio

n o

f ce

lls

Control

50ng NCS

400ng NCS

Figure 1.6.: Pulse number histograms for three selected experimental conditions.The shift of the mean of the distributions towards higher values forstronger stimulation is clearly visible, but there is also basal activityas shown by the control data set. Notably, the variability in pulsenumbers for one condition is quite large.

Control 25ng 50ng 100ng 200ng 400ng2

3

4

5

6

7

8

9

Nu

mb

er

of

pu

lses

Mean pulse number

Figure 1.7.: Mean and standard deviation of pulse numbers for every condition.The cell-to-cell variability of the number of pulses is captured bythe standard deviation.

The minimum IPI can thus only be as small as the smallest pulse width. Fortime series data where the typical IPI is much larger then the average pulsewidth, this effect is negligible. However, p53 trajectories show very broad andoften consecutive pulses with an average width of about five hours. Hence, toadequately compare IPI distributions of different experimental conditions, the

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0 10 20 30 40 50Time (h)

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IPI

Pulse peaksPulse startsPulse ends

Figure 1.8.: P53 trajectory with detected pulse starts, ends and peaks. Thenovel detection method is described in the appendix A.1. One IPIof the three IPIs present in this trajectory is exemplified. The cellwas stimulated with 25ng of NCS.

pulse width distribution was also analyzed and is shown in figure 1.10.

0 5 10 15 20 25IPI (h)

0.00

0.05

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0.15

0.20

0.25

0.30

0.35

Pro

bab

ilit

y m

ass

Control

50ng NCS

200ng NCS

400ng NCS

Figure 1.9.: IPI distributions for selected experimental conditions. The distribu-tions get broader and skewed towards longer IPIs with decreasingstimulation. This clear variation in the IPI distributions can not beexplained by the small differences in pulse widths in between thedifferent conditions as shown in figure 1.10.

The main result from the IPI analysis is, that the weaker the stimulation, the

9


more skewed is the IPI distribution towards longer IPIs. In addition, the pulsewidths are nearly the same for all conditions, with a difference in the medianbetween unstimulated and stimulated cells of one sample point or 20 minutes.This means, that the pulse appearance over time can be highly irregular anddeviates significantly from a behavior expected from a sustained oscillator forthe weak or non stimulated cells. On the contrary, the pulse shape characterizedby widths and amplitude is a robust property of the pulsatile p53 dynamics,results of their analysis are shown in figure 1.10.

(a) Median of the pulse widths forselected stimulation strengths.The median pulse width is 5.0hours for the control data set,and 4.67 hours for the otherconditions shown in the plot.

(b) Median amplitudes for selectedstimulation strengths. There isno significant variation in am-plitudes in between the differ-ent conditions.

Figure 1.10.: Analysis of pulse shapes characterized by width and amplitude.The pulse shape is essentially the same for all conditions and istherefore a robust property of the p53 dynamics.

To further illustrate the dependence of the regularity of the dynamics on thestimulation strength, box plots of the IPI distribution for all conditions are shownin figure 1.11. The IPI distribution for the cells with the strongest stimulationfairly matches the one found for the noisy limit cycle oscillator presented infigure 1.4. Consequently, oscillatory dynamics are certainly a possible dynamicalregime of the p53 dynamics, but are not sufficient to explain the dynamics ofthe weak or non stimulated cells.

The next question one may ask is about how stationary the observed dynamicsare. Meant by that is, if a certain pulsatile regime stays the same for the wholeobservation time, or if the characteristics of the pulsing changes over time.Again, the IPI distribution can serve as a measure to address this question. Thistime, only the first IPI right after stimulation and the last IPI recorded areanalyzed. If the pulsatile dynamics are stationary, the same IPI distributionsare to be expected. However, as shown in figure 1.12, for the weak to mediumstimulated cells these distributions shift to larger medians and inter quartileranges comparable to the control conditions. It follows that the pulsing activity

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Control 25ng 50ng 100ng 200ng 400ng0

5

10

15

20

IPI

(h)

Figure 1.11.: Box plots of the IPI distributions for all experimental conditions.The median and the inter quartile range of the IPIs increase withdecreasing stimulation. This indicates for non oscillatory behaviorfor the weak or non stimulated cells.

of the cells is time dependent, and is more regular right after stimulation. Takinginto account that the induced DSBs get repaired over time, a reset of the p53dynamics to basal dynamics is biologically evident. It is noteworthy that thecells stimulated with 400ng of NCS do not return to basal like dynamics evenafter 48 hours and that the unstimulated cells show a stationary behavior.

Control 25ng 200ng 400ng0

5

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IPI

(h)

first IPI

last IPI

Figure 1.12.: Box plots of the IPI distributions of the first and last IPI observedfor selected conditions to test for stationarity. The weak andmedium stimulated cells show control-like IPI distributions at theend of the observation period, indicating a return to unstimulatedbehavior.

11


1.3. Modeling theory

In this chapter some fundamental theoretic concepts used for the constructionof the p53 model shall be introduced. At first some general remarks aboutmodeling cellular processes with ordinary differential equations (ODEs) aregiven. It follows a brief discussion of negative feedback oscillators, and theirgeneral properties. These get illustrated by modeling the core negative feedbackmotif of the p53 system, as often found in the literature [34, 65]. Finally thedynamical concept of excitability is thoroughly introduced exemplified by theclassical FitzHugh-Nagumo model.

1.3.1. Preface - dynamical systems theory and biochemical reactionnetworks

The general definition of an autonomous dynamical system comprised of N statevariables x1, ..., xN is as follows

d

dtxi = Fi(x1, ..., xN ), (1.2)

that defines a system of N coupled ODEs. The time evolution of one variablexi can generally depend on the state of all other variables including xi itselfand is given by the potentially nonlinear vector-function Fi. In the frameworkof biochemical reaction networks, these variables represent concentrations ofchemical species, e.g. metabolites, enzymes or transcription factors.

Figure 1.13.: An example of a reaction network with three state variables. x1is an activator of x2, which is an inhibitor of x3. The species x3in term is an inhibitor of x1 closing the positive feedback loop. Itis also an activator of x2, therefore also establishing a negativefeedback.

Often a reaction network is presented as a directed graph, as in figure 1.13.Every direct link, xj → xi or xj � xi, between two species implicates a molecularinteraction and therefore xj actually appears inside Fi. The notion of directedlinks only makes sense, if the interactions between different state variables are

12


monotone. That means that either

∂

∂xjFi(x1, ..., xN ) > 0 for all xj > 0, (1.3)

in that case xj is an activator of xi, or

∂

∂xjFi(x1, ..., xN ) < 0 for all xj > 0. (1.4)

The last statement determines xj to be an repressor of xi.Monotonicity secures that the effect of some species xi on another species xj

always has the same direction. This property is unusual when studying generaldynamical systems theory, but quite evident when applying this theory tobiochemical reaction networks. This just means that a molecule either promotesor inhibits the production of some other molecule, it can never do both. Thisis an inherent characteristic of e.g. enzymes or transcription factors. A kinasealways promotes the formation of its phosphorylated substrate, a transcriptionfactor switches a gene either on or off and so on.

Another constraint when using dynamical systems theory to describe bio-chemical dynamics is that all state variables have to remain positive, as thereare no negative molecule numbers. This confines the state space to the positiveorthant given by O1 = {xi ≥ 0, for all i}. That further implies that the flowat the orthant boundaries is transverse, i.e. Fi(x1, ..., xn) > 0 for all xj �=i > 0.This condition assures that a trajectory started inside O1 will remain in it forall times.

A last constraint refers to the finiteness of the amount of molecules presentinside a cell. In mathematical terms this translates to the condition, that thereis an arbitrary large but finite region A ⊂ O1 for which lim

t→∞ x(t) ∈ A holds.That means that all possible solutions x(t) are bounded. In practice this is oftenassured by linear degradation terms or conservation rules.

An important concept borrowed from control theory involves the notion offeedbacks [94]. A feedback implies a closed loop in a reaction network, thatmeans there is some path connecting a species with itself, i.e. xi → xj � ... → xi.If the number of repressing interactions along this path is odd, one refers to anegative feedback, elsewise it establishes a positive feedback. Feedback mechanismsare one general concept to explain how cells can reach homeostasis even undera constantly changing environment. Remarkably, even though biomolecularnetworks can be of arbitrary complexity, a small set of feedback motifs seems tobe sufficient to explain their structure [1, 71].

All direct interactions to be modeled between molecular species xi and xj

are specified in the functions Fi and are only constricted by the monotonicity,positivity and boundedness constraints given above. However, in practice oneoften finds ODE systems solely comprised of functions like the ones given intable 1.2. The development of rate equations for chemical reactions started withsimple anorganic reactions were first and second order kinetics were sufficientto describe the observed concentration dynamics. Later in biochemistry whenenzymatic reactions became important, saturated kinetics as for the popular

13


Michaelis-Menten kinetic were developed. Including cooperativity as importantfor e.g. ligand binding led to the highly nonlinear Hill equations. For thedevelopment of all these mathematical formulations, the underlying molecularinteractions were precisely known. This is not generally true when dealing withcellular processes like transcription [11, 92] or protein degradation [12]. However,it turned out that these now phenomenological equations can still to some extendcapture the dynamics that are observed inside living cells. Saturated kineticsare a natural choice if there is some rate limiting step. This is for example thecase for transcriptional activation of a gene by a transcription factor. As thereis a finite amount of TF binding sites, the rate by which a gene is transcribedshould become independent from the TF concentration once all binding sitesare occupied. The question about how applicable a specific model is and howwell it is suited to deliver quantitative results can only be answered case-by-caseby concomitant experiments.

Table 1.2.: Overview of prominent functions used to describe biomolecular pro-cess. These terms and combinations thereof can occur as the r.h.s.of a biochemical rate equation for x. The parameter a describes abasal rate, k is the Michaelis constant and n is the Hill coefficient.

term name modeling objective ex-amples

C zero-order kinetic constant mRNA transcrip-tion

a y first order kinetic protein maturation frommRNA y

a x y mass-action kinetic degradation by y

a yy+k Michaelis-Menten kinetic transcriptional activation

by y

a yn

yn+kn Hill kinetic cooperative transcriptionalactivation by y

a 11+(y/k)n Hill repressor cooperative repression by y

1.3.2. Negative feedback oscillatorsA negative feedback oscillator is a dynamical system comprised solely of negativefeedbacks which exhibits a limit cycle regime. Such systems are deployed for awide range of biological phenomena, including circadian rhythms, cell division,gene regulation and glycolysis [38]. Models of such biochemical oscillators, e.g.the Goodwin oscillator [40], always include at least one sufficiently nonlinear

14


negative feedback motif. Such motifs are frequent in molecular biology, as meansfor autoregulation and homeostasis. A typical example is a gene activating itsown repressor, a scheme to be found for about 40% of all transcription factorsin E. coli [8].

An important subclass of such negative feedback systems are cyclic systems.These are build from pure loop structures, for which equation 1.2 simplifies to

d

dtxi = Fi(xi, xi−1). (1.5)

The condition∏N

i=1 sgn( ∂Fi∂xi−1

) = −1 assures that there is an odd numberof repressing interactions which establishes the negative feedback loop. Thefunction denoted by sgn(x) is the sign function, which gives -1 for x < 0 and+1 for x > 0. The route to oscillations for such monotone cyclic systems hasbeen theoretically investigated by many authors [44, 66, 78, 106], and the mainresults are the following:

1. There is only one stable fixed point x∗

2. Destabilization of x∗ can occur only via a Hopf bifurcation

The proofs heavily rely on the monotonicity constraint and the cyclic structuredefined in equations 1.3,1.4 and 1.5. A not overly mathematical rigorous versionof the proof can be found in the appendix of ref. [78]. The actual functionalforms of the molecular interactions formulated in the Fi’s are not important forthese results. In addition, Hopfs theorem implies the existence of a periodic orbit.The stability of that periodic orbit spawned at the bifurcation point is assuredby the boundedness of the system, stipulated in the previous section 1.3.1. Thisvery specific route to oscillations via a supercritical Hopf bifurcation involvesdistinct qualitative features of the dynamics, to be discussed and exemplified inthe following paragraphs.

Figure 1.14.: The canonical p53 autoregulatory loop. The protein P53 acts astranscription factor for the mdm2 mRNA, whereas the maturedMdm2 protein tags P53 for degradation . This constitutes amonotone cyclic negative feedback loop system.

The basic regulatory scheme of p53 introduced in section 1.1.1 fulfills exactlythe properties of a monotone cyclic negative feedback system, shown in figure 1.14.The three species involved are the p53 protein P 53 which induces transcription

15


of the Mdm2 precursor mRNA mdm2 and its matured protein Mdmd2. ThisE3-ligase tags p53 for degradation via the proteolytic pathway and closes thenegative feedback loop. Many p53 models obeying this structure have beendevised [34], here the following formulation was chosen:

d

dtP53 = C − g Mdm2

P53kmp + P53

− dP P53

d

dtmdm2 = Tm

P53kpm + P53

− dm mdm2

d

dtMdm2 = TM mdm2 − dM Mdm2.

(1.6)

The parameter C describes the constant inflow of p53 proteins given the unreg-ulated and constant transcription and translation of that gene. Degradationof the three species is given by the rates dP ,dm and dM respectively. Thismodel incorporates two saturating terms. The maximal degradation rate of P 53mediated by Mdm2 is given by the parameter g, and the maximal rate by whichP 53 can promote mdm2 transcription is limited by Tm. The Michaelis constantskmp and kpm determine the half maximum concentrations. The underlyingassumptions for using Michaelis-Menten kinetics were already discussed for thecase of transcription in the preceding section 1.3.1. The arguments mainlyrepeat in the case of the proteolytic degradation of p53. Rate limiting stepshere include e.g. the finite amount of accessible proteasomes for ubiquitinatedp53. The maturation of the Mdm2 protein is described by first order kineticswith the translation rate TM .

In accordance to the mathematical results about stability for negative feedbackloops stated above, a bifurcation analysis of this p53 system reveals a limitcycle regime bordered by two supercritical Hopf bifurcations. The bifurcationparameter, as depicted in figure 1.15, is the degradation rate dM of the Mdm2protein species. Inside the “Hopf bubble” the system undergoes sustainedoscillations, with an amplitude strongly dependent on the parameter value ofdM . The choice of this parameter is particularly reasonable given the biologicalevidence, that the main DSB sensor protein ATM directly phosphorylates Mdm2and thereby induces its autoubiquitination and degradation[70, 96].

To illustrate how such oscillators perform when changing the dynamicalregime, a time dependence of the parameter dM is introduced according to figure1.16. A fast exponential rise of the degradation rate is followed by a constantrate and eventually a slow decay. This mimics the dynamical DSB responsein a simplified way comparable to ref. [65], although no real physiologicalrelevance is actually desired here. When moving inside the limit cycle regimenegative damped dynamics are observable. These occur in the beginning mainlybecause of transient dynamics. The strong dependence of the amplitude onthe numerical value of dM implies damped oscillations when moving insidethe “Hopf bubble”. During the transition back to the steady state regimedamped oscillatory dynamics are additionally observable after passing the Hopfbifurcation point.

This damped regime is understandable by recalling, that a Hopf bifurcation is

16


0.2 0.3 0.4 0.5 0.6

dM

(h−1 )

0

1

2

3

P53 a

.u.

H1

H2

Limit cycle

Fixed point

Figure 1.15.: Bifurcation diagram of the p53 negative feedback loop model. Thebifurcation parameter dM is the degradation rate of the Mdm2protein, as defined in equation 1.6. Oscillations occur inside theHopf bubble bordered by the two Hopf bifurcations denoted by H1and H2. The amplitudes of the limit cycles are given by the lowerand upper bounds in blue and vary greatly.

defined by the crossing of the imaginary axis by a pair of conjugate eigenvaluesλ1,2 = α±iβ. This means, that close to the bifurcation point complex eigenvalueswith arbitrary small negative real part exist. They give rise to oscillatorycomponents of the trajectory when perturbed from the steady state with aperiod T ≈ 2π

β . Hence, damped oscillatory regimes exist in the vicinity of aHopf bifurcation and can not be avoided even by very fast transitions. Such aninstant transition into the “Hopf bubble” may also give rise to an overshoot,as can be seen in figure 1.17. For the bifurcation parameter dM > 0.81 theconjugation expires, the imaginary parts become zero and the real part branchesinto two distinct values. Only here the system settles down to the fixed pointwithout damped oscillations.

In summary the distinct qualitative features of the dynamics of negativefeedback oscillators are the following:

1. The amplitude of the limit cycle is strongly dependent on the bifurcationparameter

17


Figure 1.16.: Switching a negative feedback oscillator on and off. The bifurcationparameter dM is made explicitly time dependent and follows thedynamics shown in blue here. This moves the system in and out ofthe limit cycle regime, given between the marked Hopf bifurcationpoints. A (negative) damping is observable at both sides of thetransition.

2. Switching the oscillator on or off implies observable damped oscillations

3. As a corollary no isolated pulses can be generated

By recalling some features of the observed pulsatile p53 dynamics studied insection 1.2, i.e. the presence of isolated pulses as in figure 1.8 for the weakstimulated cells, negative feedback oscillators are very limited in accuratelydescribing the full dynamical range of the p53 system.

1.3.3. Introduction to excitability - case study FitzHugh Nagumo

The classical prototype of an excitable system is the FitzHugh-Nagumo (FN)model [33]. It was suggested in as early as 1961 as a 2-dimensional simplificationof the original 4-dimensional Hodgkin-Huxley model [47], which describes indetail the voltage and current dynamics of a spiking neuron. To introduce theconcept of excitability, the dynamics and bifurcations of the FN model shall bediscussed in the following.

18


(a) Numerical values for the conju-gate eigenvalue pair constitut-ing the Hopf bifurcations. Thereal part determines the stabil-ity of the fixed point, whereasthe imaginary parts cause theoscillatory components of thetrajectory.

(b) An instant transition into thelimit cycle regime may triggeran overshoot which eventuallyrelaxes on the limit cycle. Aninstant exit not too far fromthe Hopf bifurcation generallyimplicates damped oscillatorydynamics.

Figure 1.17.: (a) Eigenvalues of the fixed point for an extended range of thebifurcation parameter dM . (b) Trajectory of the system for instanttransitions in and out of the limit cycle regime.

The system is given by the equations

εdx1dt

= x1 − x31 − x2 + I

dx2dt

= γx1 − βx2 + b.

(1.7)

Here the parameter ε 1 introduces a time-scale separation, making x1 the fastand x2 the slow variable. The parameters γ,β and b are dimensionless variablesand the parameter I plays the role of an external stimulus. Because the systemis only 2-dimensional, phaseplane analysis is applicable and is sketched in figure1.18.

The cubic nullcline for the x1 variable and the linear nullcline for the x2variable intersect at the fixed point, which is stable for the excitable regime.Small perturbations from that fixed point can lead to huge excursions throughthe phasespace, as exemplified by two trajectories in figure 1.18. This behaviorstems from the strong timescale separation, here an ε = 0.05 was chosen, andcan be understood geometrically. After the perturbation the system quicklymoves horizontal till it reaches the x1 nullcline. Now the slow dynamics in x2direction are dominant until the maximum of the x1 nullcline is reached, andthe system quickly moves back to the left branch of the cubic nullcline. Fromthere it slowly settles back to steady state. The specific orbit and therefore theamplitude of such an excitation loop is dependent on the initial conditions. Such

19


Figure 1.18.: Phaseplane for the FN model in the excitability class II regime.The nullclines intersect at one point giving the stable fixed point. Asmall perturbation from that fixed point leads to a wide excursionin phasespace. Different perturbations lead to different orbits, asshown with the red and the cyan trajectory.

systems with a strong time-scale separation are also called relaxation oscillators[73]. The oscillating regime is in the FN model reached via a supercritical Hopfbifurcation, as depicted in figure 1.19. The amplitudes of the limit cycles growvery fast with the bifurcation parameter I, and the oscillatory pulses for the fastvariable x1 are very spiky. These are both consequences of the strong time-scaleseparation. It is worth mentioning, that relaxation oscillators do not belong tothe class of negative feedback oscillators.

Excitable regimes are generally close to bifurcations which result in oscillatorybehavior. The authors of ref. [49] identified two types of excitability, given by thetype of the bifurcations nearby. They are named class I and class II excitability.Relaxation oscillators generally belong to class II systems, characterized byarbitrary small amplitudes of the excitation loops and nonzero frequency at theonset of the oscillations. Interestingly, the FN model also has an excitabilityclass I regime, for which the nullclines intersect at three points as shown infigure 1.22. For that regime, no strong time-scale separation is needed, and thecorresponding parameter is relaxed to ε = 0.5.

The bifurcation scheme is a bit more complicated here, as sketched in figure1.20. An analysis for codimension one bifurcations reveals two saddle-node andtwo Hopf bifurcations. The middle fixed point is always a saddle, whereas thetwo outer fixed points are foci, which loose stability via the subcritical Hopf

20


(a) Supercritical Hopf bifurcationfor the bifurcation parameterI. The amplitudes of the limitcycle grow very quickly.

(b) The oscillatory pulses of thefast variable are very spikywhich is typical for relaxationoscillators.

Figure 1.19.: (a) Hopf bifurcation of the FN model in the excitability class IIregime. (b) Trajectories of the system in the limit cycle regime.

Figure 1.20.: Bifurcation analysis of the FN model for the parameter I withγ = 0.8. The system has three fixed points in the interval I ∈[0, 214.., 0.283..], this allwos for an excitability class I regime. Asaddle and the unstable focus are born via the two saddle-nodebifurcation points LP1 and LP2. The two outer equilibria becomestable via the two subcritical Hopf bifurcations HP1 and HP2.

bifurcations indicated by H1 and H2 in the bifurcation diagram.To extend the characterization of the system, a codimension two bifurcation

analysis with the parameters I and γ was performed. The latter parameter

21


Figure 1.21.: Bifurcation set for the two parameters I and γ. Two codimension2 bifurcations take place. The two saddle-node points collide anddisappear via the Cusp bifurcation CP1. Additionally the two Hopfbifurcations collide with the saddle-node points at the Bogdanov-Takens bifurcations BT1 and BT2. The parameter range for theexcitability class I regime ends when the two Hopf curves cross atγ = 5/6.

γ determines the slope of the x2 nullcline. As can be seen in figure 1.21 thesaddle-node points LP1 and LP2 collide with the Hopf points at the Bogdanov-Takens points BT1 and BT2. Furthermore a cusp bifurcation happens forI = 0.25 and γ = 1 where the two saddle-node curves collide and disappear.The three fixed points merge to one stable fixed point and the excitability classII regime lays right above this cusp bifurcation. The system is highly symmetricso there are actually two excitability class I regimes, they are characterized bythe co-existence of one stable fixed point, one saddle and one unstable fixedpoint. The corresponding parameter range is annotated in figure 1.21.

A saddle is a hyperbolic equilibrium with at least one negative real eigenvalue.In the two dimensional case of the FN model it is exactly one. As a corollarythere is a stable manifold which is also called separatrix. The separatrix separatesthe phase space into regions with different qualitative behavior. In the contextof excitability it serves as a direction dependent threshold. Any trajectorystarted below the separatrix will do a full excitation loop around the upperright fixed point and will eventually settle down at the stable fixed point whichis a focus in the FN model. Any trajectory started above the separatrix willundergo subthreshold damped oscillations. The phaseplane portrait in figure

22


Figure 1.22.: Phaseplane for the FN model in the excitability class I regime.The nullclines intersect at three points giving one stable focus,a saddle in the middle and an unstable focus in the upper right.Perturbations which cross the separatrix lead to an excitationloop, perturbations which do not cross the separatrix lead tosubthreshold dynamics. Parameters are I = 0.23 and γ = 0.8.

1.22 illustrates these dynamics. Perturbations leading to an excitation loopmust cross the separatrix, and therefore the threshold is direction dependent.A stable manifold of the saddle merges with the unstable manifolds from theupper unstable focus, this is called a heteroclinic connection. The excitationorbits are robust with respect to the initial conditions. That means that as longas the threshold is crossed, the amplitudes of the ensuing pulses are all verysimilar as can be seen in figure 1.23.

As stated earlier, excitable regimes are always close to bifurcations leading tolimit cycle regimes. The excitability class I regime of the FN model is close to asaddle-node on a limit cycle (LPC) bifurcation [56]. This happens when movingout of the excitable regime in positive γ direction, so right above the crossing ofthe Hopf curves in figure 1.21. The unstable limit cycles originating at the Hopfpoints collide with a stable limit cycle at the LPC points, as depicted in thebifurcation diagram in figure 1.24a.

In summary excitability class I systems exhibit excitation loops which showvery similar orbits, as long as the direction dependent threshold, the separatrix,is crossed. This carries on to the onset of oscillations, which are born with largeamplitudes and show only little dependence on the bifurcation parameter. Notime separation is needed, as the excitability stems from the specific phasespacestructure. That is the co-existence of a saddle as organizing center, one stableand one unstable fixed point. Excitability class I regimes can typically be found

23


(a) Three different excitation loops,started with different distancesfrom the separatrix.

(b) Corresponding time series forthe three excitation loopsshown in panel (a). The pulseshapes and amplitudes are verysimilar.

Figure 1.23.: As long as the threshold is crossed, different initial conditions leadto very similar orbits in excitability class I systems.

(a) Saddle-node on a limit cycle (LPC)bifurcations mark the onset of largeamplitude oscillations. The excitableregime has disappeared. This bifur-cation diagram corresponds to thehorizontal line at γ = 0.85 in figure1.21, right above the crossing of theHopf curves.

(b) Phaseplane of the FN model for pa-rameters I = 0.23 and γ = 0.85.The separatrix now separates limitcycle trajectories from trajectoriesapproaching the stable focus. Theseparatrix itself converges to a limitset, corresponding to an unstablelimit cycle, which separates the sta-ble limit cycle and co-existing stablefocus.

Figure 1.24.: Oscillating regime of the FN model right after the excitability classI regime has disappeared via LPC bifurcations.

in the vicinity of Cusp bifurcations, because they mark the appearance of three

24

1.4. The ATM-Wip1 switch building block

fixed points via pairwise saddle-node bifurcations. A bifurcation nearby leadingto oscillations is the LPC bifurcation, the emerging stable limit cycles havean arbitrary small frequency. However, in practice the period often follows arelationship like Tosc(α) ∼ Tosc + 1

αc−α , with αc being the critical point of theLPC bifurcation.

1.4. The ATM-Wip1 switch building blockAs outlined in the introductory section 1.1 the protein kinase ATM is the majorsignalling molecule of the cellular response to DSB, after activation it phospho-rylates numerous downstream effectors such as P53 directly. Phosphorylationitself is an ubiquitous post-translational modification of proteins. It can rapidlyalter the stability and the functional state of substrates. Moreover it occursoften in cascades along a signalling pathway. A prominent example is the MAPkinase pathway, where a signal coming from a receptor on the cell membraneis transmitted via a kinase cascade into the nucleus to eventually alter thetranscriptional program of the cell [21].

If kinase activity and therefore phosphorylation is a general mechanism toactivate a signalling pathway, it is evident that de-phosphorylation carried outby phosphatases is the antagonist in such a framework. For the DDR to DSBsthe phosphatase Wip1 was identified to fulfil that role [87]. Remarkably thename Wip1 stands for “Wild-type p53-induced phosphatase 1”, as it was firstcharacterized as inducted after IR in a P53 dependent manner [32]. BecauseP53 itself gets stabilized by ATM, and Wip1 dephosphorylates and thereforedeactivates ATM directly [87] a simplified feedback loop maybe written asATM → P53 → Wip1 � ATM . Therefore the kinase positively regulatesits own phosphatase and this negative feedback closes one phosphorylation-dephosphorylation cycle. This scheme is also found for other kinases, for exampleERK from the MAP kinase pathway directly activates its own phosphataseDUSP6 [100].

In the following such a generic phosphorylation-dephosphorylation systemshall be developed by the example of ATM and Wip1. The positive feedbackrequired to describe the observed ATM dynamics turned out to be sufficient toserve as the basis of an excitable p53 model.

1.4.1. ATM as Signalling SwitchATM is a large protein which consists of 3056 residues. It has many modificationand interaction domains, and it has indeed an extensive list of targets [86]. Itsfunctional interactions span a huge network, not only in cellular response toDSBs but also for example in chromatin organization and metabolism. In thefollowing the focus lies solely on the DDR response, and here it is the prominentsignalling molecule in response to DSBs [60, 82].

In unstressed conditions ATM is present as inactive homodimers in the cell.After induction of DSBs ATM gets rapidly phosphorylated and dissociates intomonomers which is the catalytic active form. It was further shown that thisphosphorylation is strongly dependent on the presence of active ATM itself,

25


which revealed that autophosphorylation takes place [4]. Active ATM thenphosphorylates the histone variant H2AX in the vicinity of a damage loci. Theseregions of γH2AX quickly expand and can elongate up to a few mega basepairs.Using immunostains of γH2AX, these sites are clearly visible as foci underthe microscope [31]. Subsequently the Mre11-Rad50-Nbs1 (MRN) complexgets recruited to the site, which is reported to enhance H2AX phosphorylationby ATM. The MRN complex is a mediator who recruits additional substratesto ATM and to the foci, it is also directly involved in DNA repair [27, 60].Moreover it is itself a substrate of the ATM kinase, establishing a positivefeedback ATM → γH2AX → MRN → ATM . The precise mechanisms ofATM activation are very complex and are still not entirely resolved, but it isclear that active ATM is required for foci formation which in turn enhancesATM activation [43, 59, 86]. To what extend complete ATM localization at thedamage loci is important for its activation is also still debated. Neverthelessit is reported that over 50% of the cellular ATM is activated within the first 5minutes after a low dose of irradiation [4]. So it appears that ATM activationacts like a switch which is dose independent and very fast.

Instead of trying to model all details of this intricate ATM activation process,the focus lies on capturing the switch dynamics rather phenomenologically. Interms of dynamical systems, a switch can be modeled by a bistable system.Bistability is long studied for chemical and biological systems, and one of themain theoretic results is, that positive feedbacks are necessary for multiplesteady states [17, 112].

A simple model that exhibits bistability may be written as

d

dtATM∗ = A

ATM∗2

ka + ATM∗2 − dAATM∗. (1.8)

Here, the variable ATM∗ denotes active monomeric ATM. The saturation of thefirst term represents various cellular factors limiting autophosphorylation andactivation of ATM. As depicted above many different molecules actually play arole in this process, all of them may be restricting. The parameter A describesthe maximal rate by which active ATM may be produced, the parameter dA

sets its lifetime.

A graphical stability analysis can be done by plotting the r.h.s. of equation1.8. The crossings of the x-axis in figure 1.25a mark three fixed points and theirlocal stability can be inferred from the sign of the r.h.s. in the respective vicinity.In the plot the dynamics around the fix points are symbolized by arrows. Thesystem exhibits two stable FPs separated by an unstable FP which serves as athreshold. As shown in figure 1.25b the system settles down to the off statewhen started below the threshold and evolves to the on state when startedabove. The threshold value tunes the sensitivity of the switch, the larger it isthe more robust is the off state against perturbations.

26


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

ATM∗

0.05

0.00

0.05

0.10

d dtATM

∗

(a) Graphical stability analysis ofthe ATM switch. The flow isindicated by arrows. The sta-ble off state is separated fromthe stable on state by an un-stable node.

0 10 20 30 40 50 60

Time a.u.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ATM

∗

(b) One dimensional ATM switchdynamics. Any trajectorystarted below the thresholdsettles down to the off state.Trajectories started above thethreshold will evolve to the onstate.

Figure 1.25.: The one dimensional ATM switch.

This description is sufficient to model ATM activation, yet the shutdown ofthe damage signal once the DSBs are repaired is of equal importance. Thedownregulation of active ATM is done by its antagonist, the phosphatase Wip1.From the modeling perspective, its action must destabilize the on state of theATM switch.

1.4.2. Incorporating the Phosphatase Wip1Studies involving mouse models describe Wip1 as an oncogene, which counteracts the tumor suppressing function of P53. Moreover overexpression of Wip1was found to be functionally equivalent to P53 inactivation and is present insome tumors, e.g. breast cancer [13]. The molecular mechanism leading to thatfunction of Wip1 were subsequently discovered. First ATM was identified to bea substrate of Wip1 in vivo, whereas depletion of Wip1 leads to an activationof ATM [87]. Detailed biochemical studies revealed many other targets of thatphosphatase, most notably for the ATM mediated DSB response are P53,Chk2,γH2AX, Rad50 and Nbs1 [114].

The dephosphorylation of γH2AX and two members of the MRN complex(Rad50 and Nbs1) directly interferes with the positive feedback ATM∗ →γH2AX → MRN → ATM∗, it is therefore assumed that Wip1 inhibits ATMactivation processes. Taking also the direct dephosphorylation of active ATMmodeled by mass action kinetics into account, Wip1 enters the equation 1.8 asfollows

d

dtATM∗ = A

ATM∗2

ka + ATM∗21

1 + W ip1kwa

− dAATM∗ − P ATM∗ Wip1. (1.9)

The parameter kwa determines the steepness of the inhibition and the parameter

27


P describes the rate at which Wip1 directly dephosphorylates ATM.To study such ATM dynamics in a simplified model, it is assumed for now that

ATM directly activates Wip1. This is an oversimplification and is solely done forsimplicity and reduction of dimensions at this stage of the model construction.The full model presented in later sections will include P53 as an transcriptionfactor for Wip1.

In the following ATM dependent Wip1 production is assumed to follow asaturated kinetic with maximal rate Tw:

d

dtWip1 = Tw

ATM∗

kpw + ATM∗ − dwWip1. (1.10)

This two dimensional system resembles a so-called bistable frustrated unit whichwas studied as a minimal model for a self activating gene with oscillatory regimes[54].

By choosing appropriate parameters, this system exhibits excitability class Idynamics. There are 3 fixed points, one stable node, a saddle and an unstablefocus. An excitation loop and the separatrix including the heteroclinic connectionto the unstable focus are shown in figure 1.26. The phasespace structure isqualitatively equivalent to the FN model in the excitability class I regime asstudied in section 1.3.3.

Figure 1.26.: Phaseplane for the simplified Atm-Wip1 model in the excitabilityclass I regime. The nullclines intersect at 3 points, giving a stablenode as the rest state, a saddle point with the incoming stablemanifold as separatrix and an unstable focus in the upper right.When the system is initialized right the threshold a huge phasespaceexcursion occurs.

28


The specific orientation of the separatrix gives an effective threshold onlywith respect to ATM∗. Increased levels of Wip1 cause a higher threshold valuein the ATM∗ direction, but can not trigger an ATM kinase pulse alone. Thismolecule specific sensibility of the ATM-Wip1 modul is shown in figure 1.27.

Figure 1.27.: A view in the phaseplane of the simplified ATM-Wip1 model shownin figure 1.26, The nullclines are omitted for better visibility. Theorientation of the separatrix makes the system unresponsive tofluctuations in Wip1. Only a perturbation in ATM∗ direction cantrigger an excitation loop.

Direction dependent thresholds, as a typical feature of excitability class Iregimes, are a natural way to reflect the robustness of a signaling network withrespect to fluctuations in specific molecule species. ATM as the primary sensormolecule for DSBs has a high sensitivity towards small concentration gainscaused by a few DSBs. However, fluctuations of the phosphatase Wip1 levelsare not prone to trigger a deceptive ATM signal pulse.

This oversimplified model captures qualitatively the excitable activation ofthe ATM damage sensor and its subsequent inactivation by Wip1. Inside livingcells, active ATM acts on the p53-Mdm2 regulatory loop. Only p53 stabilisationthen leads to transcriptional activation of Wip1. The incorporation of theseknown interactions into the model and the propagation of the ATM excitabilityshall be elucidated in the following.

29


1.5. Including the core negative feedback loop - the fullp53 model

1.5.1. The effectual modeled p53 network for the DSB response

The core negative feedback loop involving p53 and Mdm2 was already introducedin section 1.3.2, as was the damage sensor module involving ATM and Wip1 inthe preceding section. The aim of this section is to bring these two submodelstogether and thereby construct the full p53 model for the response to DSBs.

The first new interaction is the phosphorylation of the Mdm2 protein by activeATM, which was already briefly introduced when studying the bifurcations ofthe core loop in section 1.3.2. The authors of ref. [96] showed that Mdm2 getsrapidly destabilized by DSB-dependent PI 3-kinase family members, where ATMbelongs to. Phosphorylation and subsequent auto-ubiquitination followed byproteasomal degradation is the main mechanism for that. Hence, ATM∗ entersthe core negative feedback loop by promoting Mdm2 degradation. A simple massaction kinetics term was chosen to model that interaction. A second influence ofATM∗ on the core loop is its reported weakening of the Mdm2 - P 53 interaction.The p53 protein itself is a target of ATM∗ and Mdm2 has a lower binding affinityto phosphorylated P 53. As with the present data phosphorylated p53 can not bequantified, there is no additional species introduced into the model. Therefore,this mechanism is indirectly captured rather phenomenologically by inhibitingthe Mdm2 dependent P53 degradation via ATM∗ . The relative strength ofthis second ATM∗ interaction is given by the parameter R in the model. Thesecond extension so far concerns the transcriptional activation of Wip1 by p53.Transcription and subsequent translation and maturation of proteins happentypically on a time-scale of hours, as opposed to (de-)phosphorylation andubiquitination which typically occur within minutes. By adding an explicit Wip1mRNA species to the equations, as it was also done for Mdm2, these differenttime-scales get represented in the model without introducing explicit delays inthe equations. Finally the initial trigger of the ATM activation are the DSBs,they enter the equation for ATM∗ via the signal function S ≡ S(DSB) to bespecified later.

With these modifications and extensions the now complete p53 model is givenby the following equations:

30

1.5. Including the core negative feedback loop - the full p53 model

d

dtATM∗ = A

ATM∗2

kA + ATM∗21

1 + Wip1/kW A− dAATM∗ − P ATM∗ Wip1 + S(DSB)

d

dtP53 = C − dP P53 − g Mdm2

P53kMP + P53

(1 +

R

1 + ATM∗

)d

dtmdm2 = Tm

P53kP m + P53

− dm mdm2

d

dtMdm2 = TM mdm2 − dM Mdm2 − dAM ATM∗ Mdm2

d

dtwip1 = Tw

P53kP w + P53

− dw wip1

d

dtWip1 = TW wip1 − dW Wip1

(1.11)

A graphical scheme of the modeled interaction network is shown in figure 1.28.The only new parameters are the halflife of the Wip1 mRNA dw, the rate by whichthe Wip1 protein maturates TW and the rate by which active ATM promotes theMdm2 degradation dAM . All other parameters were already introduced in therespective sections 1.3.2 and 1.4. The naming of the parameters follows thesegeneral conventions: parameters containing the letter “T” denote productionrates concerning translation or transcription, parameters starting with a “d”denote degradation rates and parameters containing “k” are the respectiveMichaelis constants. The subscripts encode the affiliation to the modeled species,e.g. a “m” in the subscript is associated with the Mdm2 mRNA and a “M”with the respective protein species. Thus the parameter kP m, for example,describes the Michaelis constant of the Mdm2 mRNA production inducedby the p53 protein. Rate parameters which do not follow this nomenclatureare: A which gives the maximal activation rate of ATM, P which describesthe dephosphorylation of ATM∗ by Wip1, g which gives the maximal Mdm2dependent degradation of P53 and finally C which is the maturation rate ofnew P53 entering the system. An overview of all model parameters and theirvalues is given in the appendix A.2.

31


p53

ATM*P

Wip1

Mdm2

mdm2

P53

Ub

wip1

dP

DSBsP

dP

P

Figure 1.28.: Schematical overview of the modeled p53 interaction network in re-sponse to DSBs. A P at an interaction stands for phosphorylation,an Ub for ubiquitination and the dP marks dephosphorylations.The protein species involved are in yellow, magenta denotes mR-NAs. The ATM kinase activation is triggered by DSBs, which enterthe equation via the signal function S(DSB) defined in the maintext. Active ATM molecules facilitate the formation of furtheractive ATM molecules via (auto-)phosphorylation events. Thesetarget the E3-ligase Mdm2 for phosphorylation which gets in turndestabilized by autoubiquitination and subsequent degradation.P53 gets constantly transcribed and translated, this steady influxis marked by the dashed arrow pointing from the p53 promoter.Because the antagonist Mdm2 got depleted, the P53 levels riseand the transcriptional activation of its targets Wip1 and Mdm2is amplified. When the concentration of the matured phosphataseWip1 increases, the direct dephosphorylation of ATM and espe-cially the interference with the positive feedback leads to a declineof the active ATM concentration. Mdm2 gets in turn no longersuppressed and eventually brings p53 back to the steady stateconcentration. This completes one pulse cycle. If there are enoughDSBs left a second pulse will evolve accordingly.

32


1.5.2. Bifurcation analysis and deterministic dynamicsThe positive feedback introduced to capture the ATM-Wip1 switch dynamicshas the potential to also cause bistability and excitability in the complete p53network considered here. And indeed by choosing appropriate parameters, aregime with one stable rest state, a saddle point and an unstable fixed pointcan be found and is shown in figure 1.29. The value of the signal functionS ≡ S(DSB) was chosen as bifurcation parameter. As it should be for no DSBspresent, i.e. S = 0, the system resides in an excitable class I regime characterizedby one stable and two unstable fixed points.

Figure 1.29.: Bifurcation diagram for the full model with the value of the signalfunction S to be the free parameter. The system has for S = 0 threefixed points, a stable rest state, a saddle point and an unstablefixed point. This fulfills the properties of an excitable class Isystem.

Before looking at the model dynamics, the bifurcation analysis shall beextended to other parameters and the region of excitability in the parameterspaceshall be identified. It turns out, that apart for S there are only 4 types ofbifurcation diagrams as depicted in figure 1.30.

The excitability class I regime is generally to be found when there are 3 fixedpoints present, two of them have to be unstable as depicted by the dashed curves.The size of this region is fairly large for all parameters shown in figure 1.30.The types I and II show symmetry in the sense of their asymptotic behavior.By that is meant, that for example transforming the fixed point curve F forthe parameter C with Ftr(C) = F ( 1

C ), the resulting diagram is qualitativelythe same as the one for the parameter g in subfigure 1.30b. The types III andIV also show a symmetry with respect to a Ftr = F (−C) transformation. Anoverview for all model parameters is given in table 1.3.

33


(a) Type I (b) Type II

(c) Type III (d) Type IV

Figure 1.30.: The four types of bifurcation diagrams obtained for the full modeland exemplified by the parameters denoted on the x axes. Theexcitability class I regime is between the saddle-node bifurcationpoints LP2 and LP1. If the Hopf point HP1 lays in between thesaddle node points, then there is a small region of bistability andthe excitable regime ends there. The parameterized input signal isS = 0.01

Interestingly, there is a straightforward characterization for all types by notingthe direct effect of their respective parameters on P 53 or ATM∗ . Types I andII contain all parameters controlling the P53-Mdm2 negative feedback loop,whereas types III and IV contain all parameters controlling the ATM-Wip1switch. Although a nondimensionalization shall not be carried out for the model,mainly to preserve the biological meaning of the parameters, it can be assumedthat parameters belonging to the same type would likely be condensed by it.

To further extend the bifurcation analysis, a search for codimension-2 bifurca-tions was performed as shown in figure 1.31. In accordance with the analysisof the FN model in section 1.3.3, a Cusp bifurcation point is found. This

34


Table 1.3.: Overview of the bifurcation diagram type membership for all modelparameters

type I type II type III type IV

Parameters C, dm, dM

dAM , kmp

g, Tm, TM

RP, Tw, TW A, dw, dW

kW A, kP w

Trait positive onP53

negative onP53

negative onATM∗

positive onATM∗

marks the appearance of the phasespace structure required for the excitabilityclass I regime. There is additionally a BT point, spawning a Hopf bifurcationcurve which effectively destabilizes the upper fixed point and makes the systemexcitable. The specific bifurcation parameters are C and S here. However, thereare actually only 2 symmetric types of these codimension-2 diagrams, the othertype can be found in the appendix A.4. The region of excitability extends farinto the halfplane for S < 0, but the signal should always be constraint to S ≥ 0so this is omitted in this plot.

Figure 1.31.: Bifurcation set for the full p53 model, parameters are C and S.The excitability class I regime lays in between the two saddlenode curves, this is the region were 3 fixed points co-exist. Thedestabiliziation of the 3rd fixed point occurs via a subcritical Hopfbifurcation, so that the region of excitability is additionally confinedby the Hopf curve. This curve is spawned at the Bogdanov-Takenspoint BT1, which is very close to the Cusp point CP1.

35


For a wider picture of dynamical regimes present in the model, a bifurcationset with an extended parameter range is shown in figure 1.32. An oscillatoryregime is nearby, this is a feature of excitable systems in general as discussed insection 1.3.3. But there are also bistable and monostable regimes. To obtainthe bifurcation diagrams shown in figures 1.29 and 1.30, one just has to drawan imaginary horizontal or vertical line through the bifurcation set.

Figure 1.32.: Overview of dynamical regimes present in the p53 model. Thep53 DNA damage response is modeled by exploiting the excitableand oscillatory regimes. The Bogdanov-Takens point spawning theHopf curve is omitted for better visibility.

The bifurcation leading to the oscillations in the p53 model is the saddle-node homoclinic bifurcation [56]. This is a global bifurcation involving a localbifurcation. What happens at the bifurcation point is, that the heteroclinicconnection of the saddle to the stable fixed point becomes a homoclinic orbitof the merged saddle-node. This saddle-node then disappears via the localsaddle-node bifurcation and a limit cycle appears near the former homoclinicorbit. From within the excitable regime, which is considered as operating pointof the model for no DSBs present, the oscillatory regime can only be reachedby increasing the signal S. This is shown in figure 1.33, which is an extendedversion of figure 1.29.

The limit cycles of this positive feedback oscillator are born with huge ampli-tudes, and show only very little dependence on the signal strength up to S ≈ 0.5.In effect, for no or low signal the system resides in an excitable state capable ofshowing isolated pulses. In addition, for a stronger signal after the saddle-nodehomoclinic bifurcation the system undergoes stable sustained oscillations. These

36


Figure 1.33.: Appearance of a stable limit cycle in the p53 model. The LPC2points mark the saddle-node homoclinic bifurcation. The amplitudeof the oscillations is nearly constant up to S ≈ 0.5. The productionrate of p53 is set to C = 1.4.

are exactly the characteristics found in the p53 single cell data for low vs. highdamage input, which were discussed in section 1.2. The actual functional formof S ≡ S(DSBs(t)) has to qualitatively reflect these dynamics depending on thenumber of DSBs present. This is discussed in the next section. For now whatremains is to simulate some actual deterministic trajectories of the p53 model.

In figure 1.34 the system was first excited by a small initial pulse of ATM∗

at t = 0, all other variables were initialized on steady state levels. Then afterthis excitation pulse the system settles down to the stable rest state. At t = 15the signal is instantly set to S = 0.4. This moves the system in the limit cycleregime, and the oscillator is switched on. From the maximal stimulation anexponential decay of the signal takes place, and the system leaves the limit cycleregime at t ≈ 33. This qualitatively resembles an experiment, in which the cellsare damaged at t = 15. It is a feature of such positive feedback oscillators tofinish the last cycle undamped. The reason for that is, that the orbit of the limitcycle smoothly transforms into the excitation loop at the homoclinic bifurcationpoint. It is noteworthy, that the last pulse is a bit delayed. This is an effectsimilar to that one briefly discussed in section 1.3.3, namely that the periodof the oscillator tends to infinity close to the bifurcation point. However, inpractice this region often is negligible small in parameter space, more detailsabout that can be found in the appendix A.5. Finally, the oscillator has beenswitched off and after that the system remains silent as there is no further input.This undamped switching behavior is qualitatively very different in comparisonto pure negative feedback oscillators. Their switching behavior was alreadystudied in figure 1.16 of section 1.3.2 and it shows that damped pulses cangenerally not be avoided when moving in or out of the oscillatory regime in such

37


Figure 1.34.: Deterministic excitable and oscillatory dynamics of the p53 model.At t = 0 the system is initialized above the threshold and undergoesan excitation loop. At t = 15, after it settled down to the stablerest state, the signal is instantaneous set to S = 0.4. This switchesthe positive feedback oscillator on. The signal then exponentiallydecays and at t ≈ 33 the oscillator is switched off. The system stillundergoes a complete last cycle before returning to the rest state.

systems.In summary the system shows a fairly large region of excitability in parame-

terspace. With increasing signal strength, the system becomes oscillatory via asaddle-node homoclinic bifurcation. Both regimes have been identified in thedata in section 1.2, i.e. excitable dynamics for low or no external damage signaland oscillatory dynamics for the strong stimulated cells. For the latter scenario,the system responds in a digital fashion, i.e. with a discrete number of completepulses. These oscillatory pulses are almost indistinguishable from the excitoryones, which is in agreement with the uniform pulse shapes found in the data. Inthe next section, the forcing of the system via an explicit time dependent signalS(DSB(t)) shall be studied with respect to a simple stochastic model for theDSB dynamics.

38

1.6. Driving the p53 model with a stochastic DSB process

1.6. Driving the p53 model with a stochastic DSBprocess

1.6.1. Constructing a stochastic process for the DSB dynamics

The dynamics of the DSBs are considered as the primary input of the p53 model.Fortunately, as outlined in section 1.1, single cell trajectories of damage foci areavailable. While the direct relation between foci number and number of DSBsremains unclear, they can still serve as a quantitative marker [63]. Therefore, thenumber of foci is treated as the number of DSBs. Typically the DSB dynamicsare described by an exponential model [51], which sufficiently describes therepair process. However, the half-lifes of the foci show large variability across acell population [51, 63].

At first a pool of single cell foci trajectories is analyzed, the same raw datawas already used for ref. [63]. Here the emphasis also lies on the amount ofDSBs present after the repair of the inflicted damage has finished, this amountis referred to as background damage level. A modified exponential decay

DSBdet(t) = N0 exp−λt + Nb, (1.12)

is fitted to the data. The background damage level is captured by Nb, N0 is theinitial amount of DSBs and 1/λ is the decay rate. To capture the exponentialrepair process as well as the background level, a dataset with moderate damageof 5Gy and an observation time of 24h was chosen. Two examples are shown infigure 1.35.

(a) N0 = 103.58, λ = 0.29, Nb =5.15

(b) N0 = 53.59, λ = 0.40, Nb =3.86

Figure 1.35.: Two representive single cell trajectories of the foci number for cellsstimulated with 5Gy. The modified exponential fits are shown asdashed curves. The fitting parameters according to equation 1.12are shown below. After an exponential decay of the foci, a meanbackground damage level, captured by the fitting parameter Nb,remains.

39


Repeating the fitting for the whole cell population yields a fitting parameterdistribution shown in figure 1.36.

(a) Distribution of the inverse ofthe decay rate λ.

(b) Distribution of the damagebackground Nb.

Figure 1.36.: Fitting parameter distributions for the modified exponential modelof equation 1.12 describing the foci dynamics. Cells were damagedwith 5Gy and tracked for 24 hours in this data set. The meanfitting parameter values are shown in the legends.

As reported in the literature, a broad distribution of decay times rangingfrom 1-7 hours is found. This indicates a large cell-to-cell variability in theeffective DSB repair times. The mean number of damage loci as captured bythe modified exponential repair model is 〈Nb〉 ≈ 2.3.

However, in another data set where the number of foci in undamaged cellswas recorded for 12 hours, the mean number of foci in the population is around〈Nb〉 ≈ 8.4. The decay visible in figure 1.37 is due to photo bleaching effects.Given the unstressed conditions, the variability in the foci number within onetrajectory over time could be calculated. Quantified by the variance (Var),also this quantity varies a lot between cells and ranges approximately from7 < V ar < 70.

In summary, exact quantification of the DSB dynamics from experimentaldata remains a challenging task. What can be inferred from the raw dataavailable is the following:

1. Upon damage induction, an exponential decay of the number of DSBs isobserved. The decay rates show a large cell-to-cell variability,

2. After the repair of the induced damage is finished, there is a persistentnumber of DSBs present inside the cells also without stimulation,

3. The mean of this background damage is of the order 2 < 〈Nb〉 < 10,

4. This quantity significantly fluctuates over time

Given these characteristics, a stochastic process describing the DSB dynamicsshall be constructed in the following. The sources of the observed stochasticity

40


(a) Three exemplary single cell foci tra-jectories taken from a dataset ofunstimulated cells. The number offoci variates significantly over time.

(b) Median and quartiles of the focinumber for an unstimulated popula-tion of cells. The population meanfor the background damage level is〈Nb〉 ≈ 8.4

Figure 1.37.: Single cell foci dynamics and population dynamics of unstimulatedcells.

of the DSB dynamics are both the processes which cause the DNA damage aswell as the cellular repair mechanisms. The former include e.g. the errors arisingfrom mitotic division, the occurrence of radical metabolic byproducts and cosmicradiation, which are intrinsically noisy. The latter appear to be irregular alsodue to various reasons, e.g. different severity of the DNA lesions or different copynumbers and spatial availability of the proteins involved in the repair process.As shown in figure 1.37a and as quantified by the variance, the trajectoriesshow indeed a variability atypical for a deterministic process. The stochasticprocess describing the DSB dynamics should be of discrete state, as the numberof DSBs is integer valued, and homogeneous in time for both computationaland analytical tractability. A well studied and powerful theoretical frameworkproviding these features concerns Markovian birth-death processes [37]. Underthe rather weak assumption, that at a specific instant of time there is only eitherone DSB occurring or one is being repaired, it can be readily applied to modelthe DSB process.

At first, two rates b and r are defined by the following assignments of proba-bilities, given that there are n DSBs present at time t:

b dt = probability that a new DSB occurs in [t, t + dt]nr dt = probability that a DSB is repaired in [t, t + dt].

(1.13)

The evolution of the DSB process DSB(t) is now governed by the following

41


probabilities:

P(

DSB(t + dt) = m|DSB(t) = n)

=

⎧⎪⎨⎪⎩

bdt, if m = n + 1nrdt, if m = n − 11 − (b + nr)dt, if m = n

.

(1.14)

Following to the reasoning of Gillespie in ref. [37] this process is called thepayroll process and the quantity (b + nr)dt describes the probability that theprocess will jump away from state n in the next infinitesimal time interval[t, t + dt]. Denoting the probability that the process will not leave that state inthe time interval [t, t + τ ] by P0(τ) and the laws of probability give:

P0(τ + dτ) = P0(τ) [1 − (b + nr)dτ ], (1.15)

which implies a differential equation with the solution

P0(τ) = e−(b+nr)τ . (1.16)

Combining this result with the definitions in equation 1.13 and noting thatunconditionally leaving a state at τ +dτ and the landing at a new state m = n+νare independent events yields the next-jump density function given by:

ρDSB

(τ, ν|n, t

)=

⎧⎪⎨⎪⎩

b e−(b+nr)τ , if ν = +1 and n ≥ 0nr e−(b+nr)τ , if ν = −1 and n ≥ 00, otherwise

. (1.17)

This density can be fed into the Gillespie algorithm, and providing an initialcondition, DSB(0) = N0, Monte Carlo simulations of the stochastic DSB processcan be readily computed. The stationary distribution of this process is thePoisson distribution given by the following density function:

ρs(n) =(b/r)k e−b/r

n!(1.18)

The asymptotic analytical solutions for the mean and variance read:

limt→∞ 〈DSB(t)〉 = Nb =

b

r, and

limt→∞ V ar

(DSB(t)

)=

b

r.

(1.19)

These relations only provide the ratio between the two rates b and r. Toeffectively reverse calculate the rates from the foci data, dynamical propertiesof the process have to be taken into account. The aim here is not to claim orprovide an exact parameter estimation, but to give a reasonable estimation giventhe simplistic stochastic model used and the data available. In the following,the mean foci number Nb is used for the estimation, this already gives b = Nbr.The first moment time-evolution function for the payroll process given in ref.

42


[37] reads:

〈DSB〉t = e−rt(

N0 +∫ t

0bert′

dt′)

=(

N0 − b

r

)e−rt +

b

r.

(1.20)

Substituting with the asymptotic mean yields

〈DSB〉t = (N0 − Nb)e−rt + Nb. (1.21)

The latter formulation remarkably resembles the modified exponential repairmodel introduced in equation 1.12. The notion of Nb and 〈Nb〉 might be alittle confusing. For the stochastic process, these two are identical, Nb beingthe asymptotic time average of one realization and 〈Nb〉 being the asymptoticensemble average. For the time series foci data, these two hardly coincide.Reasons for that are the finite and rather short period of sampling, someadditional cell-to-cell variability and probably some systematic error in themeasurements, as the number of foci is only a proxy for the number of DSBs.Nevertheless, to advance with the estimation, the l.h.s. of equation 1.21 istreated as an ensemble average at time t. Solving for the repair rate one obtains:

r = t−1ln

(N0 − Nb

〈DSB〉t − Nb

)(1.22)

The time dependence of the rate is obviously artificial, as the rate should betime independent for a time homogeneous process. And indeed, when usingthis formula to reverse calculate the repair rate out of an ensemble of syntheticdata, r is time independent for all 0 < t < tc. An example is shown in figure1.38. However, when using equation 1.22, one carefully has to observe thedenominator. From a critical time on, the ensemble average at time tc gets veryclose to the asymptotic average: 〈DSB〉tc

≈ Nb. This quickly leads to numericalprecision issues, as the denominator converges exponentially fast to zero andmight even turn negative due to fluctuations. This effect is clearly visible infigure 1.38b. Practically this means, that the estimation of the repair rate rshould be carried out in a time domain, where the trajectories on average arestill decaying. To improve the estimate, an averaging within the respective timedomain [0, tc] can be carried out and will be done for the inference from thereal data. It is noteworthy, that the relaxation dynamics are needed for theparameter estimation. An unstimulated ensemble of trajectories correspondingto stationary dynamics alone would not allow for the described method as thiswould correspond to tc = 0.

Having demonstrated that the proposed method for rate parameter estimationworks with synthetic data, now the real data shall be analyzed accordingly.Because every foci trajectory has generally a different N0, the data is binnedto form subensembles with comparable initial amount of DSBs. As can beseen in figure 1.39, the initial amount of DSBs present in the cells has a broaddistribution despite the population received a fixed damage dose. This makes

43


(a) Two ensembles and their ensem-ble averages of the stochasticDSB process. The estimationis based on 300 realisations foreach starting value.

(b) Estimation of the parameterr using equation 1.22 of themain text and the observa-tions shown in the left panel.The critical times upon numer-ical problems arise are approx-imately tc = 400 and tc = 600respectively.

Figure 1.38.: Demonstration of the method for reverse calculating the repairrate given only observations of the stochastic DSB process. Thesynthetic data was computed using the Gillespie algorithm, param-eters are r = 2, b = 5. By estimating the asymptotic mean andusing equation 1.19 all parameters of the stochastic process canbe inferred from observations.

the parameter estimation even more difficult, due to the low sample numbers inthe subensembles.

As there are no principle problems in applying the method for the parameterestimation, the results for two different subensembles for the 5Gy data set areshown in figure 1.40. As expected, the results look rather poor as the chosenensembles for the 5Gy set only contain 17 and 15 cells respectively. Given thebroad initial distribution and that there are only 63 cells in total, there is nosatisfactory pooling available. But nevertheless, the repair rate was estimatedto be in the order of r ≈ 0.325 per hour, which yields an average lifetime of adamage locus to be around 3 hours. Given the mean damage background levelto be 〈Nb〉 ≈ 2.3, the corresponding spontaneous break rate for the DSB processis b ≈ 0.8. Although a similar analysis for the 10Gy set gives comparable valuesfor the repair rate, this data set is not considered as suitable for the estimationprocess. As most of the trajectories have not assumed a stationary dynamic,the estimation of the background damage level is too vague.

As stated earlier, this by no means constitutes a satisfying or exact parameterestimation. The author believes to have captured the right order of magnitude,as it is the same as the mean of the foci-lifetimes extracted from the data in ref.[63]. But the exact values may still differ substantially from this analysis. But

44


(a) Distribution of the initialamount of DSBs in the 5Gydataset. The distribution ap-pears to be bimodal.

(b) Distribution of the initialamount of DSBs in the 10Gydataset.

Figure 1.39.: Initial DSB distributions for the 5Gy and 10Gy single cell focitrajectories.

instead of picking some rates by educated guessing, a method was found andapplied to in principle reliably extract the needed information from the data,given the stochastic process defined above.

Figure 1.40.: Repair rate estimation for the 5Gy foci data set and based onthe Markovian DSB model. The number of samples in the twoensembles considered is 17 and 15 respectively. This may explainat least in part the rather poor performance of the estimationperformance. Nevertheless, the resulting value of r ≈ 0.35 has areasonable order of magnitude.

In summary, a Markovian birth-death process was formulated to model theDSB dynamics. A parameter estimation method based on relaxation dynamicsof such processes was constructed, and successfully applied to synthetic data. Itwas further applied to the foci trajectories best suited for the estimation process,

45


and reasonable parameter values were obtained. In the following section thisprocess shall be applied to the p53 model.

1.6.2. Stochastic forcing of the excitable p53 model

The general idea pursued here is, that a high number of DSBs translates intoa high signal strength S(DSB(t)) which according to the bifurcation analysisin section 1.5 cause an oscillatory behavior of the model. As the parameters ofthe stochastic DSB process have been estimated in the preceding section andits exponential relaxation dynamics are in good accordance to the foci data,damaging the cells is simulated by simply starting the process away from itsasymptotic mean with a N0 > Nb. On the contrary, the spontaneous pulsesobserved in the data and treated as excitation loops in the model may capturedby the stationary dynamics of the stochastic DSB process, independent ofexternal stimulation. The fluctuations around the mean background damagemay cause a spontaneous p53 pulse if the resulting signal S(DSB(t)) crossesthe excitation threshold of the model. Hence, the specific functional form of Sis crucial for the model performance. On the other hand, the exact molecularmechanism that triggers an initial amount of active ATM before the positivefeedback kicks in is not known. In summary there remain the following twodegrees of freedom: the threshold for the excitation loop can be adjusted byvarying the parameters of the model and the function S can be chosen freelyto cross this threshold for any DSB number desired. As the parameter setchosen in section 1.5 already qualitatively reflects p53 pulse properties like shapeand lengths, what remains here is to make the signal function S explicit. Theanalysis is generally restricted to qualitatively show, that the model capturesthe p53 dynamics of both stimulated and unstimulated cells.

By forcing the system of ODEs with the DSB process it becomes non-autonomous, as it strictly already was in section 1.5. There, S ≡ S(t) was madeexplicitly time dependent to illustrate the switching dynamics of the positivefeedback oscillator. As the DSB process itself is stochastic, the p53 modelbecomes a hybrid stochastic-deterministic model. When simulating the system,it is convenient to precompute the DSB trajectory, as their is no feedback fromthe deterministic p53 model to this stochastic process. Upon the integrationprocess, the precomputed realisation of the DSB process DSBr(t) can now betreated like a deterministic forcing function.

Before fixing the function S(DSB(t)), the 5Gy foci data set of the precedingsection shall be reconsidered, this time also looking at the concomitantly recordedp53 dynamics. For that, the data is grouped in two mutually complementingsets. In the first set the data gets binned with respect to the initial damagereceived, measured by N0. Then the distribution of pulse numbers is evaluatedfor the cell in each bin. And secondly the data is grouped according to thenumber of p53 pulses a single cell trajectory exhibited, and the ensemble averageof the corresponding foci trajectories is plotted. The results are shown in figure1.41, and clearly show, that cells with more initial damage and slower repairdynamics tend to have more pulses. This restates the results from the literatureand also from the data analysis of section 1.2, but here the quantification is

46


more exact given the available foci data.

(a) Distribution of pulse numbersfor the cells binned accordingto the initial amount of foci N0.The higher the initial damage,the higher the average pulsenumber.

(b) Foci trajectory ensemble aver-ages, grouped by the numberof p53 pulses the cells exhib-ited. The difference betweenone and two, and three andfour pulses, is on average moreinfluenced by the repair kinet-ics than by the initial damagereceived.

Figure 1.41.: Relation between foci trajectories and p53 pulsing for the 5Gydata set.

The signal function S(DSB(t)) could be of many forms, and there is no apriori precedence for one or the other. For simplicity a scaled logarithm waschosen :

S(

DSB(t))

= γ ln(

DSB(t) + 1)

. (1.23)

The single parameter γ controls the sensitivity of the signal towards DSBs andis therefore very essential for the model performance. For example, setting γ tohigh would not allow for the excitable regime even for stationary DSB dynamicsas the resulting value S(Nb) would already set the system in the limit cycleregime. Given the mean number of pulses to be around one per 12 hours forthe dataset analyzed in section 1.2, the sensitivity parameter was adjusted toγ = 0.06. The estimated rate parameters for the stochastic DSB process arefixed to b = 0.8 and r = 0.3, all what is needed to simulate a trajectory isto specify an initial amount of damage N0. Some exemplary realisations areplotted in figure 1.42.

By simulating pools of trajectories, model output statistics for different initialconditions can be obtained. This facilitates the assessment if the model semi-quantitatively captures the measured data. In figure 1.43 the IPI distributionsand the pulse number distributions for simulated and measured data are shown.The IPI distributions are, as discussed in section 1.2.1, a suitable measure, todemonstrate that the model depending on the initial damage exhibits bothregular oscillatory behavior and irregular excitable behavior. This corresponds

47


(a) (b)

(c) (d)

Figure 1.42.: Four exemplary p53 model trajectories. In figures (a) and (b) theinitial amount of DSBs was set to N0 = 100 which correspondsto a high damage scenario. Figures (c) and (d) show two tra-jectories initialized with N0 = 2. These dynamics correspond tounstimulated cells.

to the strong and weak stimulated cells in the data respectively. The pulsenumber distributions illustrate the high variability in the total pulse numberin between single cells for an equal amount of initial DSBs. In the model, theonly source for this variability is the stochastic DSB process but it is sufficientto generate a heterogeneity similar to that found in the data. However, asshown in the preceding subsection, the broad distribution of damage loci givena fixed stimulation strength additionally amplifies the heterogeneity in the data.Another limiting factor for the congruence between simulations and real datais measurement noise, which is not assessed at all by the model. Recordingfluorescence signals from living cells introduces many sources of variability in thesignal, even when the tagged protein levels are constant. The measured pulsestherefore show some irregularities, whereas the simulated pulses are always ofequal shape. This effects, that the IPI distributions of the data are broadenedeven for the most regular pulsing cells similar to the effects observed for thenoisy limit cycle oscillator analyzed in section 1.2.1.

48


(a) (b)

(c) (d)

Figure 1.43.: Model output statistics (left) compared to the results from thedata analysis (right). In figure (a) and (b) the IPI distributionsare shown. Figures (c) and (d) show the pulse number distribution.For the simulated data, the initial amount of damage was fixedaccording to the legend. Whereas in the measured data, a fixeddose of the damaging agent NCS was added. Both simulatedquantities are in good agreement with the data.

In this section the p53 model for the response to DSBs is finally complete.The signaling function was heuristically chosen in a way, that only one additionalparameter controls its sensitivity towards the amount of damage. Taking thebasal p53 activity as a benchmark, this parameter was fixed and model outputstatistics were generated. These showed good agreement with the data accordingto the IPI and pulse statistics. In the last section, the predictive power of thismodel shall be assessed.

49


1.7. Reanalysis of inhibitor experimentsThe kinase inhibitor Wortmannin (Wm) was used in previous works [7, 62] tostudy the effects of transient activation of the p53 network. Wortmannin is abroad-range kinase inhibitor effectively abrogating the activities of many kinases,notably the upstream kinases ATM and DNA-PK. In these studies, Wm wasadded shortly after the cells were damaged and single cell pulse analysis wasperformed. The authors found a reduced number of pulsing cells dependent onthe time point of Wm addition and stated an all-or-none response with respectto amplitude and widths of the p53 pulses. Consequentially it were these studies,which first introduced the concept of excitability to explain the p53 dynamicsin response to DSBs.

However, by disabling ATM the p53 network considered here it the modellooses its only positive feedback. The theoretical considerations undertakenin section 1.3 therefore render an excitable behavior of the model downstreamof ATM as impossible, given that there is no other positive feedback on p53.Moreover, simulations show a graded dependence of pulse amplitude and widthon the time point of Wm addition, results are shown in figure 1.44. By loweringthe amount of active ATM within a pulse formation process, Mdm2 levels canrecover sooner and P53 is set back to the steady state concentration before afull pulse can evolve. To address the question if the model maybe still lacks aprominent feedback mechanism, the raw data used in ref. [62] shall be reanalysed.In these experiments, the cells were damaged with a high dosage of NCS andsubsequently imaged for six hours. Wm was added at different time points foreach condition and thereafter refreshed to continuously inactivate the upstreamkinases.

An important part of pulse detection for the p53 data involves a threshold forthe minimum amplitude a pulse should have to be identified as such. Also thewavelet based method used for the analysis here can be deluded by some randomfluorescence signal variations. The fraction of responding cells, the are ones whoshow a pulse, as a function of the amplitude threshold is shown in figure 1.45.As expected the number of detected pulses decreases with increasing amplitudethreshold. The question about the right threshold is hard to answer exactly.Therefore the subsequent analysis will be done for three representative thresholdvalues. In the results of ref. [62] a response rate of around 0.6 is reported forthe control condition. This suggests, that a very high threshold was chosen intheir analysis.

To make the effects of Wm on the pulse amplitudes for different thresholds andalso to the results of the model comparable, the pulse amplitude of the controlcondition is set to one. With this, the amplitudes of the conditions with inhibitoraddition are given as the ratio of the amplitude of the unperturbed p53 system.In figure 1.46 the results of the data analysis for three different thresholds andthe model output are shown. The earlier the inhibitor is added the lower arethe p53 pulse amplitudes to be observed. This effect gets smaller with higherthresholds for the pulse detection, as this effectively filters out the smaller pulses.In this sense, with higher thresholds the analysis here converges to the resultsof ref. [62]. As already seen in figure 1.44 the model qualitatively shows exactly

50

1.7. Reanalysis of inhibitor experiments

Figure 1.44.: Deterministic simulations of the inhibitor experiments. Dashedlines correspond to Wortmannin addition after 60 minutes, point-dashed lines after 15 minutes, of stimulation. The system wasstarted above the excitation threshold. Upon the indicated timepoints, a strong degradation term mimicking the effects of Wmwas switched on in the r.h.s. of the ATM equation of the p53model. The model predicts lower pulse amplitudes and smallerwidths compared to the control condition.

this behavior, although the effect is more pronounced. Reasons for that mightbe that there are some intermediate and redundant kinase species acting onthe P53-Mdm2 core negative feedback which are not covered by this minimalp53 model. These may longer suppress the P53 antagonist Mdm2 and thereforebuffer the sudden absence of active ATM in time, effectively adding inertnessto the systems dynamics. Additionally, the kinetics of the kinase inhibition byWm are unknown. In the model, as can be seen in figure 1.44, its inhibitingaction kicks in instantly after addition. Lowering this rate of inhibition wouldtrivially lead to bigger p53 pulses in the model. The same analysis for the pulsewidths yields comparable results. However, given the low time resolution inthis data set of only 25 time points for the whole observational period, thisanalysis has more uncertainties and is therefore omitted here, As a last remarkit should be stated here, that the analysis was supported by visual inspectionof single trajectories to double check that detected low amplitude pulses areindeed present in the data and not artefacts of the detection algorithm.

The observation of figure 1.45, that the addition of the kinase inhibitor Wmlowers the response rate of a stimulated cell population in a time dependentmanner is not captured by the model simulations. This can be understood bynoting, that the timing of the first pulse is very heterogeneous within a cellpopulation. This is shown in figure 1.47, where the distribution of the peaktimes is shown. It is striking that the timing of the detected pulses is shifted

51


Figure 1.45.: The fraction of cells to be identified as responding as a function ofthe amplitude threshold used for the pulse detection. The additionof the inhibitor Wm generally decreases the number of cells showinga pulse. The earlier Wm is added, the more cells show no p53pulse at all. A comparison to the results reported in ref. [62] show,that a very high threshold was used in their analysis.

towards earlier time points the earlier Wm is added. It is evident that theinitiations of the individual pulse formations shift accordingly. But this meansthat compared to the control condition, the potentially later forming pulses aremuch stronger inhibited or completely abrogated by the kinase inhibitor. Theseare surely contributing to the lower response rates. As the cell state is modeledcompletely homogeneous there is no cell-to-cell variability present in the modelto account for the observed different pulse timings. This explains at least inpart that the lower population response rates upon Wm addition can not becaptured by simulations so far.

In this section the predictions of the model led to a careful reanalysis ofpublished data. In accordance with the simulations, the inhibitor Wm indeedhas an influence on the pulse amplitude. This is a strong check for modelconsistency. If the reanalysis would not have shown such effects, the assumedunderlying regulatory network would have been rendered incomplete, as thiswould be a strong evidence for another upstream kinase independent positivefeedback present in the system to predominantly account for the excitability.Therefore, these results assure, that the positive feedback employed in the modelis indeed important for the excitable pulse formation.

52

1.7. Reanalysis of inhibitor experiments

Figure 1.46.: Comparison of median amplitudes extracted from the data fordifferent time points of Wm addition with model predictions fordifferent detection thresholds. The earlier the inhibitor Wm isadded, the lower are the resulting p53 pulses. This effect getssmaller with higher thresholds. The model qualitatively reflects thisbehavior, although the effect of the inhibitor is more pronouncedin the simulations.

Figure 1.47.: Distribution of the p53 pulse peak timing for strongly stimulatedcells. The kinase inhibitor Wm is added as indicated in the legend.The earlier it is added, the more of the detected pulse peak timesare shifted towards earlier time points. Cells in which the pulseformation started later will not develop a pulse depending on thetime point of Wm addition. This explains at least in part the lowerresponse rates observed in figure 1.45.

53


1.8. Discussing the p53 modeling approach

Every approach to quantitatively describe cellular processes faces the immensecomplexity of the molecular interactions constituting a specific cellular function.This is still puzzling for both the theoretical as well as the experimental biologist.It is therefore of utmost importance to define and confine the biological problemas much as possible to keep the complexity as low as possible. One can argue thatespecially for p53 this idea is doomed to fail, as the number of feedbacks involvedwith p53 are of the order 100 [14]. Even when only considering the specificregulation of p53 in response to DSBs, the amount of potentially importantinteractions is still hard to anticipate [64].

It was therefore very advantageous for the model construction that experi-mental findings, especially the ones documented in refs. [7, 62, 87, 96], providedstrong clues, that the four protein species P 53, Mdm2, Wip1 and ATM are thekey players for orchestrating the signalling cascade in response to DSBs. Incor-porating the respective mRNA species which are actively regulated during thatresponse, a p53 model consisting of only six species could be developed whichsemi-quantitatively captures the observed p53 dynamics for both stimulated andunstimulated cells. In contrast to most published models also the basal dynamicswhich are characterized by isolated pulses can be reproduced. The model alsoshows a digital response (see figure 1.34 in section 1.5.2) after damage induction,reported already for the first single cell studies in ref. [57]. The theoreticalconcept which allows for that complex model behavior is excitability, which isa well studied subject of dynamical systems theory. A detailed introductionto excitability based on the famous FitzHugh-Nagumo model was developedin section 1.3.3. Applying this theory to molecular interaction networks leadto the insight, that only systems comprised of at least one positive feedbackcan exhibit an excitable regime. Models prior to this work either did not in-corporate a positive feedback at all [6, 35, 65], or relied on positive feedbackswhich lack a strong experimental evidence [15, 97]. Besides, the concept ofexcitability was still not explicitly exploited by the latter. A careful scan ofpotential positive feedbacks was done in close collaboration with the Loewer labwhich also provided transcriptome analysis of some promising candidates, e.g.Caspase-2 [75]. This lead to the conclusion, that the positive feedback is mostlikely to be found in the upstream kinases, most notably ATM. The reportedswitch like activation of ATM, as it is very sensitive towards low numbers ofDSBs, made it a prominent candidate. However, its specific interactions withother kinases like DNA-Pks, the MRN repair complex or the phosphorylatedhistone variant γH2AX leading to the positive feedback remain elusive. Furtherexperimental studies are very eligible here. In the end only a phenomenologicalself activation term for the ATM activation was added to the model, which ispresumably an oversimplification. Incorporating the phosphatase Wip1 whicheffectively destabilizes active ATM and interferes with the positive feedback wassufficient to introduce excitability into system (see section 1.4).

The model construction was generally supported by extensive single cellraw data analysis (section 1.2). To reliably detect and characterize the p53pulses a novel peak detection algorithm was developed (see Appendix A.1). It

54


exploits the favorable properties of wavelet analysis of noisy data. In short, bysmoothing the data on many scales the typical data analysis problem of over-or under-smoothing is circumvented. The distribution of inter pulse intervalswas found to reliably identify dynamical regimes which deviate from noisy limitcycle oscillators. Applying this tools to measured p53 trajectories lead to theconclusion that both oscillatory and excitable regimes are present in the data.The former are most prominent in strong stimulated cells whereas the latterare found in weak or unstimulated cells. Even for a fixed and high amountof damage, the pulsatile responses are very heterogeneous with respect to thenumber of pulses on a single cell level.

This observed heterogeneity could be further assessed by analysing singlecell damage foci trajectories. Similar to the results of ref. [63], a high cell-to-cell variability of the DSB repair kinetics has been observed. However,special attention has been given here to the background damage levels. Thesefluctuating DSB dynamics after repair or without stimulation at all serve as theinput for the basal p53 dynamics. A simple yet plausible stochastic process forthe DSB dynamics was proposed to be a Markovian birth and death process(section 1.6.1). It captures the two essential events, repair of a single DSB andoccurrence of a new DSB, with two rates. Despite the quantitative uncertaintiesin the experimental assessment and discrepancies between different data sets,reasonable values for the rates of the stochastic DSB process could be estimatedfrom the data. A promising extension of the stochastic process devised here,would be to make the rates time dependent. By varying the repair rate, differentrepair processes dominant in different cell cycle phases as reported in ref. [51]could be captured. A time dependent break rate could directly reflect theincrease of breaks naturally occurring during DNA replication in the S phaseof the cell cycle. However, these alterations would make the stochastic processnon-homogeneous in time which is beyond the scope of this work. The conversionof the number of DSBs present and their effective signal strength S(DSB(t))for the excitable p53 model has been achieved by an ad hoc one parametriclogarithmic transformation.

The proximity of an oscillatory regime in parameter space is a hallmark ofevery excitable regime. This is naturally exploited in the model, as it is thesignal strength which determines the transition between these two regimes (seesection 1.5.2). Therefore the number of DSBs determines weather p53 oscillatesor shows isolated infrequent pulses. The time an individual cell spends in theoscillatory regime is given by the stochastic repair dynamics incorporated inthe Markovian DSB process. Therefore the same amount of initial damagecan lead to different pulse number responses as observed for the measured p53trajectories (section 1.6.2).

Despite the simplifications and assumptions made during model construction,it proved to improve the mechanistic understanding of the p53 signaling modul.Simulations but also just theoretical reasoning pointed to the existence of smallerand partial p53 pulses upon ATM inhibition some time after stimulation. Drivenby the model predictions, published p53 trajectories obtained from inhibitorexperiments were carefully reanalyzed and such lower amplitude pulses wereindeed discovered (section 1.7). Thus, data analysis and modelling suggest that

55


there is no excitability, e.g. all or none responses, downstream of the kinasenetwork targeted by the kinase inhibitor Wortmannin. This hypothesis couldbe tested experimentally in future studies.

Having all sources of cell-to-cell variability reside in the stochastic DSBdynamics is most likely the major limitation of the model. It is well known thatprotein abundances as well as responsiveness to uniform stimuli differ widelybetween cells in a clonal population [95]. One major source of this variability isstochastic gene expression [29]. Fluctuations around the steady state proteinlevels are especially functionally important for low abundances. However, studieslike the one in ref. [85] showed that the key players considered here, namely p53,Mdm2, Wip1 and ATM, are all highly abundant in mammalian cells. This, andalso the observation of rather smooth and regular p53 pulses, greatly reduces thepossible impact of stochastic gene expression on the p53 system. This shouldnot be that surprising, given the crucial role p53 plays in securing the genomicintegrity and cell cycle progression. A fact which might play an important roleis that also the levels of highly abundant proteins are variable within a cellpopulation and they can have long, i.e. several cell cycles, mixing times [88].On the contrary, the steady state concentrations and also all the production anddegradation rates are the same for all “cells” described by the model. Anotherpotential source of variability in cell responses given the same number of DSBsis the spatial distribution of the damage loci. The exact processes triggering thefast and global ATM activation are not well understood so far. If nucleationprocesses play an important role, the location of the damage loci and thespatial availability of damage sensors and other mediators may be important.In summary, for the present model a fixed trajectory of DSBs, deterministicallytranslated into the signal strength S(t), will trigger identical system responsesfor all simulated cells. This is needless to say not observed in experiments[63]. Including a meaningful cell-to-cell variability in the model downstream ofthe DSB process is a challenging task for the future. It will ultimately effectthe excitation threshold and could thereby explain the observed variability inresponsiveness.

An interesting finding which might be of more general significance concerns theswitching behavior of biological limit cycle oscillators. Oscillatory dynamics arereported for a broad range of cellular processes including metabolism, signaling,locomotion or cell division. For many of these oscillatory processes it can beexpected that the oscillations are not running forever, but instead that they are awell regulated response towards a changing environment. For example oscillatorysignaling might be initiated or terminated by extracellular stimuli, or metabolicoscillations might depend on the available nutrients. By observing the onset orthe termination of such oscillations, results from dynamical systems theory allowfor a qualitative inference of the regulatory network which accounts for theseoscillations. The route to limit cycle oscillations for a negative feedback systemalways implies strong damping in amplitude during the switching between anoscillatory regime and a steady state. On the contrary, for a positive feedbackoscillator the oscillations are generally born with huge amplitudes. The reasonfor that distinct qualitative behavior is the type of bifurcation leading to theoscillations. For negative feedback oscillators this is the supercritical Hopf

56


bifurcation, which directly yields stable oscillations of arbitrary small amplitudes.In the case of the positive feedback system the route to oscillations is morecomplicated. A subcritical Hopf bifurcation with a subsequent bifurcation oflimit cycles is one way to reach stable oscillations. Another possibility is thesaddle-node homoclinic bifurcation, where the saddle collides with a stable limitcycle. Excitability class I regimes are most likely to be found in the proximity ofsuch bifurcations. Also, strictly speaking, positive feedbacks are only necessaryfor these type of oscillators, they do not guarantee such complex bifurcationstructure. However, reversing this argument implies that pure negative feedbacksystems are incapable of excitable dynamics and switch oscillations on andoff always with varying amplitudes. This was the main argument for theconclusion that the known feedbacks of p53 involving Mdm2, Wip1 and ATMare insufficient to explain the systems behavior. It should be noted, that whenomitting the switching behavior and no isolated pulses are observed, the popularnegative feedback oscillator models certainly match the observations of sustainedoscillations and are comparatively simple to employ. The future prospectivewould be to survey other cellular oscillators and check their established regulatorynetwork for consistency in this regard, e.g. the transforming growth factor βmight be a promising candidate [2, 110].

57

2. Hierarchic stochastic modelling ofintracellular Ca2+

So far, an intracellular signaling system focused on p53 was modeled by anunderlying deterministic dynamical system exhibiting regular limit cycle behaviorand irregular excitable behavior driven by noisy DSB dynamics. The observedpulsatile dynamics indicated a regulatory network comprised of at least onestrong positive feedback, which precise molecular basis still has to be found. Inthe second chapter of this work another important intracellular signaling system,focused on the messenger molecule Ca2+ , shall be investigated. Interestingly, italso exhibits irregular pulsatile, although more spiky, dynamics. However, thepositive feedback allowing for such sharp excitable dynamics is well known inthe case of Ca2+ signaling. It is termed calcium induced calcium release (CICR)mechanism and will be explained in the following introductory section. Becausethe Ca2+ spikes themselves are very short compared to a typical inter spikeinterval, the system can be modeled adequately by focusing on the stochasticspike occurrence times. In contrast, for p53 the IPIs were of the same orderof magnitude as the p53 pulse width. Additionally, the recorded single cellCa2+ trajectories generally show a very high degree of variability. Thereforeall mechanistic processes are captured by probabilities and the Ca2+ signalingmodel to be discussed is completely stochastic.

The results and findings presented here are based on recently published work[72, 105] by the Falcke group. Experimental results shown were done by co-authors Kevin Thurley from the Falcke group and Stephen C. Tovey and AbhaMeena from the Colin W. Taylor lab at the University of Cambridge.

2.1. Introduction to intracellular Ca2+ signalingCa2+ is a ubiquitous intracellular messenger transmitting information by repeti-tive cytosolic concentration spikes. It is involved in key cellular functions likeproliferation, metabolism and apoptosis, as well as in cell type specific functionslike muscle contraction or insulin secretion [9, 30]. The intracellular Ca2+

dynamics can be captured by life cell imaging using fluorescent calcium sensitivedyes [9, 30]. A typical outcome of such an experiment are single cell Ca2+ spiketrains, a few examples are shown in figure 2.1. The main descriptor of thesespike trains is the interspike interval (ISI) distribution. The ISI was originallydefined in the exact same way as the IPI is defined in the case of p53 signalsdiscussed in the first part of this work. It is simply the time period betweentwo consecutive Ca2+ spikes. It has been demonstrated by analysis of the ISIdistribution that these Ca2+ signals are repetitive stochastic events [26, 90].The main argument here is, that the average ISI (Tav ) is of the same order of

59

2. Hierarchic stochastic modelling of intracellular Ca2+

magnitude as its standard deviation σ. Moreover, a large cell-to-cell variabilityof Tav is observed. Strikingly, the relationship between Tav and σ is linear, andits slope is a robust cell type specific property [90, 103].

Figure 2.1.: Schematic overview of the Ca2+ release mechanism, a detaileddescription can be found in the main text. Additionally threeexemplary Ca2+ spike trains are shown. Analysis of the interspikeinterval distribution revealed the stochasticity of these intracellularCa2+ signals. Figure taken from ref. [105]

An important class of Ca2+ signals is mediated by Inositol-1,4,5-trisphosphate(IP3), whose production is facilitated by G protein coupled cell surface receptors[9, 30]. Upon binding of an extracellular ligand to the receptor, phospholipaseC (PLC) gets activated and cleaves membrane phospholipids which yields IP3 .This then binds IP3 receptors (IP3Rs) in the endoplasmatic reticulum (ER) andthus sensitizes them for activation by Ca2+. Active IP3Rs act as Ca2+ channels,releasing Ca2+ ions from the ER lumen into the cytosol. Sarco-endoplasmaticreticulum Ca2+ ATPases pump Ca2+ back into the ER after release. Thetransient increases in cytosolic [Ca2+] trigger downstream effects like activationof protein kinase C [9, 74]. IP3Rs are organized as clusters of about 1 to 20 IP3Rmolecules [68, 91, 98]. A schematic overview of the release mechanism is shownin figure 2.1. Upon sensitization by IP3, the clusters are successively activatedby Ca2+-induced Ca2+ release (CICR). This mechanism is based on the openingprobability of IP3Rs, which increases with the local Ca2+ concentration, up toa threshold value where further increase of the Ca2+ concentration becomesinhibitory [10, 99]. The clustering of IP3Rs implies that cellular Ca2+ signalsresult from a hierarchic cascade of single channel opening (’blips’) over clusteropening (’puffs’) to opening of several clusters (’wave’ or ’spike’). Stochasticity isreported for all of these events [67]. Thus, the Ca2+ spikes arise by a multiscale

60

2.2. An analytical approach to hierarchic stochastic modelling

stochastic process emerging by clustering of IP3 receptors.An approach to model such complex stochastic systems was coined hierarchic

stochastic modeling (HSM), its theoretical developments started already someyears ago [61, 79]. The successful application to Ca2+ dynamics was achievedrecently [103] and further analytical insights were gained even more recently[72] and shall be presented in the following.

2.2. An analytical approach to hierarchic stochasticmodelling

In this section the general theoretical framework involved in hierarchic stochasticmodeling (HSM) shall be developed. At first, the general idea to refrain froma pure Markovian description shall be motivated. The formal consequencesincluding semi-Markovian processes, the correspondent non-Markovian Masterequations and the concept of probability fluxes will be discussed subsequently.Finally, a specific Ca2+ model developed in ref. [103] shall be analytically solvedin the context of a first passage time problem.

2.2.1. What is HSM ?

The main goal of HSM is a state space reduction without fully neglecting themicroscopic dynamics of the system. As to be seen in the following, this effectivelyimplies a semi-Markovian description. However, the greater theoretical challengeis paid off well by a substantial reduction in the number of free parameters inthe model. Additionally, it makes the theory readily applicable to experiments,which especially in molecular biology rarely observe microscopic state changesdirectly. This is due to the enormous complexity of actual molecular interactionsoften found for elementary cellular processes such as transcription, translation orCa2+ signaling. The many cooperative interactions translate via combinatoricsto high dimensional state spaces which in turn make the application of standardmethods like the chemical master equation often intractable. The integrationof many microscopic states into one mesoscopic observable state is therefore anaturally choice for the description of intracellular processes and is the core ideaof HSM.

When describing a receptor channel molecule, often the main question ofinterest is if the channel is open or closed. The receptor molecule might havemany internal states where only one corresponds to the channel being open[22, 93]. However, in a standard Markovian description, all internal statetransitions have to be described, also the ones which are not leading to anopening event. The classical master equation reads

∂

∂tP j

i (t) =N∑

l=1

[qliP

jl (t) − qilP

ji (t)

]. (2.1)

Here, N is the number of system states and P ji (t) = P (i, t|j, 0) is the conditional

probability that the system is in state i at time t conditioned by being in state

61


j at time t = 0. The qil are the rates for the microscopic state transitions i → l.As an example the state space of a single hypothetical channel molecule with fourinternal states is shown in figure 2.2. The system becomes quickly intractableif one considers multiple, say Nch copies, of such a receptor as the state spacedimension grows by a power law N ∼ 4Nch . However, only transitions to or fromstate x4 change the functional state of the channel, all other internal transitionsdo not trigger an opening or closing event. Thus, on a higher systems levelwhere these internal dynamics are of no explicit interest, an integration of allmicroscopic states corresponding to the observable closed state is reasonable.Together with a corresponding open state this effectively reduces the number ofstates per receptor to two, S1 and S2. Given that realistic channel models oftenconsider eight or more internal states [22], this greatly reduces the state spacedimension, which now grows according to N ∼ 2Nch .

trans

ition

pro

b.

time

Figure 2.2.: (A) State space of a hypothetic Ca2+ channel with four microscopicstates and associated Markovian transition rates qi,j . The channelis considered as open only for state x4 and closed for all otherstates. (B) Sketch of the HSM ansatz which combines many mi-croscopic states xi to fewer functional observable states Si. (C)The transitions between this functional states are modeled by non-exponential waiting time densities Ψi,j in contrast to the exponentialwaiting times with rates given by the qi,j describing the Markoviantransitions between microscopic states. Figure taken from ref. [72]

In a standard coarse-grained stochastic systems description, simple rates wouldbe assigned to the transitions between S1 and S2 to preserve the Markovianproperties of the model. This would, however, completely neglect the underlyingmicroscopic dynamics. A more realistic approach is to consider explicitly non-

62


exponential waiting times Ψi,j for the observable state transitions, as thesestill carry information about the underlying microscopic state transitions. Inthe following, the corresponding mathematical framework called semi-Markovprocesses shall be introduced. The concept of ordinary Markov processes shallalso be briefly discussed in that context.

2.2.2. Semi-Markov processes

The stochastic process of consideration X(Tn) has the finite sample spaceΩ = {0, 1, ..., N} consisting of all states the process can visit. The Tn ∈ R

+ formthe sequence of transition times between these states with T0 < T1 < ... < Tn.Now transition probabilities according to

Qi,j(τ) = P{Xn+1 = j, Tn+1 − Tn ≤ τ |Xn = i} (2.2)

can be formulated. The process is temporally homogeneous by noting theindependence of Qi,j(τ) from n, and it is also assumed Qi,j(0) = 0.

A process which is governed by Eq. 2.2 is called a semi-Markov process [16,20, 83], with Q = {Qi,j(τ); i, j ∈ Ω, τ ∈ R

+} forming a so called semi-Markoviankernel. The reasoning behind that terminology will be briefly summarized. Bydefining

pi,j = limτ→∞ Qi,j(τ) (2.3)

one gets the transition probabilities for the i → j transition of the embeddedMarkov chain with normalization condition

∑Nij pi,j = 1. In case of a pure

Markov process with unconditioned transition rates qi,j

pi,j =qi,j∑N

k,i�=j qi,k

(2.4)

holds. Now one can define the following probability distributions:

Gi,j(τ) =Qi,j(τ)

pi,j= P{Tn+1 − Tn ≤ τ |Xn = i, Xn+1 = j}. (2.5)

The successive visits of the process Xn form a Markov chain with transitionprobabilities pi,j , whereas the length of the sojourn time intervals [Tn, Tn+1) aregiven by the distribution functions Gi,j(τ). If these distributions can be writtenas Gi,j(τ) = 1 − e−

∑k

qi,kτ ≡ Gi(τ), than the process is a pure Markov processwith rates qi,j , the sojourn times are exponentially distributed and independentof the next state. In that case the process is memoryless, which means that

P{Tn+1 − Tn > s + t|Tn+1 > t} = P{Tn+1 − Tn > s}, (2.6)

for s, t > 0 applies.In conclusion, a semi-Markov process (X, Tn) still fulfills the Markov property

with respect to the subsequent state transitions, but allows arbitrary (withrespect to

∑j lim

τ→∞ Qi,j(τ) = 1) sojourn time distributions and hence looses itsmemorylessness with respect to the transition times Tn+1 − Tn. It is therefore

63


the ideal framework to apply the desired non-exponential waiting times used bythe HSM approach, now exactly defined by:

d

dτQi,j(τ)dτ = Ψi,jdτ = P{t < Tn+1 − Tn < τ + dτ |Xn+1 = j, Xn = i}. (2.7)

Therefore, the term conditioned waiting time seems appropriate. By notingthat the transition probabilities of the embedded Markov chain are given bypi,j =

∫∞0 Ψi,j(τ)dt, the semi-Markov process is completely defined by a set of

conditioned waiting time densities. Also note that by definition of the transitionprobabilities in Eq. 2.2, the time variable τ does not correspond to a systemstime, but describes the wait after the last transition Tn.

2.2.3. Non-Markovian master equations and first passage times

By substituting the Markovian rates qi,j with the semi-Markovian waiting timedensities Ψi,j to describe the state transitions i → j, the conventional Masterequation is no longer applicable. It is replaced by a more general formulation:

dPi,j(t)dt

=N∑

l �=j

Iil,j(t) −

N∑l �=j

Iij,l(t), (2.8)

which employs the concept of probability fluxes. Iil,j(t) denotes the probability

flux for the transition from state l to j under the condition that the processstarted at state i at time T0 = 0. Following the arguments of ref. [79] thesefluxes are in turn recursively determined by the waiting time densities:

Iil,j(t) =

∫ t

0Ψl,j(t − τ)

Nin∑k

Iik,l(τ)dτ + f i

l,j(t). (2.9)

This is a convolution of the conditioned waiting time to go from state j to l andall incoming fluxes to the state j. The f i

l,j are the initial fluxes with f il,j(t) ≡ 0

for i �= l. Plugging this equation into the non-Markovian Master equation 2.8yields an integro-differential equation for the probabilities Pi,j(t). Because theintegral equation for the fluxes constitutes a convolution kernel, a solution bythe Laplace Transform L{f(t)} =

∫∞0 e−stf(t)dt = f(s) seems promising as

L{f(t) ∗ g(t)} = L{f(t)}L{g(t)} (2.10)

holds. This effectively turns equation 2.9 into an algebraic problem for theLaplace transformed fluxes Ii

l,j , which can be solved by standard methods givena transition structure defined by a set of Ψi,j ’s. The application to a specificCa2+ spiking model is the subject of the next subsection.

An important quantity for stochastic processes with many applications is thefirst passage time (FPT), given by a FPT density Fi,j(t). It assigns a probabilitythat the stochastic process started in state i at time t = 0 will first visit state jat time t. The state j often constitutes a threshold or is special by other means,so it is of interest to assess its FPT distribution. An elegant way of obtaining an

64


expression for the FPT is the renewal equation [107]. It formulates the generalidentity that the probability to start in state i at time 0 and to be in state j attime t with i �= j is equal to the probability to arrive there for the first time atany time between 0 and t and to recur there at time t

Pi,j(t) =∫ t

0dτFi,j(t − τ)Pj,j(τ). (2.11)

This constitutes an integral equation with convolution kernel for Fi,j(t), by usingequation 2.10 the algebraic equation for the Laplace transformed FPT reads

Fi,j(s) =Pi,j(s)Pj,j(s)

. (2.12)

As the Laplace transform is also used to solve for the probability fluxes, whatremains is to transform the non-Markovian Master equation 2.8:

sPi,j(s) − δij =∑

l

Iil,j(s) −

∑l

Iij,l(s). (2.13)

Putting everything together one obtains a formula for the Laplace transformedFPT density:

Fi,j(s) =∑

l Iil,j(s) −∑

l Iij,l(s)∑

l Ijl,j(s) − ∑

l Ijj,l(s) + 1

. (2.14)

Together with equation 2.9, a set of waiting time densities Ψi,j completelydetermines the FPT problem as an algebraic problem in Laplace space. The bigadvantage of this approach emerges by noting that

− ∂

∂sf(s)

∣∣∣s=0

=∫ ∞

0t f(t)dt, (2.15)

which equals the first moment of t when f(t) is a probability density. Bysuccessive application of the derivative with respect to s one obtains for themoments of the FPT density:

〈tn〉 = (−1)n ∂n

∂snFi,j(s)

∣∣∣s=0

. (2.16)

Hence, it is possible to calculate arbitrary moments from the Laplace transformedFPT density Fi,j without facing the often cumbersome back transformation tothe time domain.

Before directly applying this analytical method to a complex real life example,namely stochastic Ca2+ signaling, it shall be demonstrated on a small examplesystem in the next section. There, also the conceptual differences betweenMarkovian and semi-Markovian system descriptions shall be briefly discussed.

2.2.4. A simple semi-Markovian system

In this section, the developed analytic method to solve the FPT problem forsemi-Markovian systems shall be demonstrated on a small example system.

65


Additionally, some considerations about the equivalence of semi-Markovian andMarkovian systems in the context of FPTs shall be developed. The systemunder consideration is a linear chain comprised of only three states. Therefore,the system exhibits a simple transition structure shown in figure 2.3.

0 1 2

Figure 2.3.: Schematic state space and transition overview of a small examplesystem, which only has three states and constitutes a linear chain. Isis used to demonstrate the developed first passage time formalism inthe preceding section. If this system is Markovian or semi-Markoviandepends on the definitions of the conditioned waiting times Ψi,j .Both cases will be studied in the main text.

Without fixing the conditioned waiting times Ψi,j , the transition structureimposed in figure 2.3 is sufficient to derive a general solution of the FPT problemto go from state 0 to state 2. Equation 2.14 of the preceding section gives theLaplace transformed FPT density in terms of the Laplace transformed fluxes.The fluxes in turn are given by the solution of equation 2.9, which in Laplacespace is a matrix equation:

Ij = ΨIj + f j , (2.17)

basic algebra yieldsf j = (� − Ψ)Ij . (2.18)

Here Ij denotes the vector of the probability fluxes Ij = (Ij01, Ij

10, Ij12, Ij

21) andthe f j are the initial function vectors f0 = (Ψ01, 0, 0, 0) and f2 = (0, 0, 0, Ψ21).The superscript denotes the respective initial state of the system, where both thestarting (0) and the destination (2) state are needed given the renewal approachin equation 2.11. Finally Ψ is the matrix of conditioned waiting times, which iscompletely determined by the transition structure of the simple example system:

Ψ =

⎛⎜⎜⎜⎝

0 Ψ01 0 0Ψ10 0 0 Ψ10Ψ12 0 0 Ψ120 0 Ψ21 0

⎞⎟⎟⎟⎠ .

66


The matrix equation 2.18 is an inhomogeneous linear system of equations forthe Laplace transformed fluxes Iil which can be solved exactly by standardalgebraic methods. The solution yields all Laplace transformed fluxes as algebraicexpressions of the Laplace transformed waiting times. Plugging these in equation2.14 gives the general solution of the FPT problem in Laplace space:

F0,2(s) =Ψ01Ψ12

1 − Ψ01Ψ10. (2.19)

Following equation 2.16 the mean first passage time is given by:

〈t〉0,2 = − ∂

∂sF0,2(s)

∣∣∣s=0

. (2.20)

After differentiation and defining Υ = 1 − Ψ01Ψ10 one obtains:

〈t〉0,2 = − 1Υ

[(Ψ01Ψ′

01Ψ10Υ

+ Ψ′01

)Ψ12 +

(Ψ01Ψ12Ψ′

10Υ

+ Ψ′12

)Ψ01

] ∣∣∣∣∣s=0

,

(2.21)where the variable s was suppressed for better readability. This equation onlycontains expressions of the form Ψi,j(0) and Ψ′

i,j(0) which have a straightforwardinterpretation. By using the Laplace transformation property shown in equation2.15 and the definitions of the preceding section about semi-Markovian systems(equations 2.5 and 2.7) one obtains:

Ψi,j(0) =∫ ∞

0Ψi,j(t)dt = pi,j

Ψ′i,j(0) = −

∫ ∞

0t Ψi,j(t)dt = −pi,jTi,j . (2.22)

Here the pi,j are the familiar transition probabilities of the embedded Markovchain and the Ti,j are the individual mean waiting times to go from state i tothe adjacent state j. Applying these relations to equation 2.21 finally gives:

〈t〉semi0,2 =

1p12

(T01 + p10T10 + p12T12) . (2.23)

Hence, the mean FPT only depends on the mean individual waiting times Ti,j

and the transition probabilities pi,j . This general solution, i.e. no specific Ψi,j ’shave been used yet, allows to compute the mean FPT without the need for asingle explicit Laplace transformation. It additionally paves the way to assessthe equivalence of Markovian and semi-Markovian systems with respect to FPTs.

To introduce a Markovian systems description, rates are assigned for eachtransition:

0 λ−−⇀↽−−γ

1 α−−⇀↽−−γ

2

67


The theory of continuous time Markov chains [37] states that the unconditionedwaiting time densities are the following: Φ0(t) = λe−λt, Φ1(t) = (α + γ)e−(α+γ)t

and Φ2(t) = γe−γt. By considering the transition probabilities of the embedded(discrete time) Markov chain, namely p01 = 1, p10 = γ

α+γ , p12 = αα+γ and

p21 = 1, we can define the Markovian conditioned waiting times according tothe definitions of section 2.2.2:

Ψ01 = p01Φ0 = λe−λt

Ψ10 = p10Φ1 = γe(α+γ)t

Ψ12 = p12Φ1 = αe(α+γ)t

Ψ21 = p21Φ2 = γe−γt. (2.24)

The individual mean waiting times are T0,1 = 1/λ, T1,0 = T1,2 = 1/(α + γ)and T2,1 = 1/γ. Noting the symmetry T1,0 = T1,2 simplifies equation 2.23 evenfurther, therefore the solution of the Markovian FPT problem is:

〈t〉Markov0,2 =

1p1,2

(T0,1 + T1,2) =λ + α + γ

λα(2.25)

A striking feature of Markovian systems is that the waiting time distributionsare the same for all possible transitions from a specific state i. This symmetryimplies for the conditioned waiting times

ΨMarkovi,j (t) ≡ pi,jΦi(t) = qie

−∑

jqi,jt

, (2.26)

where the sum goes over all directly adjacent states. The Markovian waitingtimes are given by Φi =

∑j qi,je−

∑qi,jt and the transition probabilities by

pi,j = qi/∑

j qi,j . Moreover, for Markovian systems the transition probabilitiesand the individual waiting times are not independent, but are both determinedby the rates qi,j . On the contrary, for semi-Markovian systems, transitionprobabilities and transition times are completely independent:

Ψsemii,j (t) ≡ pi,jΦi,j(t). (2.27)

Here the Φi,j ’s are arbitrary probability density functions on the non-negativereals and the transition probabilities can be freely chosen as long as

∑j pi,j = 1

is fulfilled. This has far-reaching consequences, e.g. highly probable transitionscan be arbitrarily slow and vice versa. In the spirit of HSM, it is interesting tothink about how to (re-)construct the large Markovian (microscopic) system,which, with the right state space partitioning, operates a specific semi-Markovian(observable) systems dynamic. Although, this is beyond the scope of this work.

Finally, Markovian and semi-Markovian systems with the same transitionstructure are equivalent with respect to the mean FPT, if the transition proba-bilities and the mean individual waiting times are the same for both systems. Inparticular that means that Ti,j = Ti must hold for the semi-Markovian system,which is a symmetry Markovian systems naturally obey. Although this doesnot necessarily impose a symmetry on the semi-Markovian waiting times (i.e.

68


Φi,j ≡ Φi), as only their first moment must coincide. It is straightforwardto extend this argumentation to the nth moment of the FPT, as this wouldlead to an expression for <tn> containing all derivatives of the waiting times:[Ψi,j(0), Ψ′

i,j(0), ..., Ψni,j(0)]. Hence, if also higher moments of the individual

waiting times coincide, there is still equivalence. Although in practice, alreadythe variances (2nd moment) of the semi-Markovian Φ′

i,js will generally differfrom the variances of the Markovian exponential waiting times, which are simplygiven by T 2

i,j = 1/(∑

qi,j)2.Having now assembled and demonstrated all theoretical tools needed for an

application of HSM to Ca2+ signaling, a specific analytically treatable Ca2+

model shall be analysed in the following. The non-exponential waiting timesdefined in the context of semi-Markov processes will be employed to describe theintracellular stochastic Ca2+ release process. The inter spike interval statisticswill be treated as a FPT problem, which solution strategy was outlined in theprevious section.

2.2.5. Explicit solutions for the tetrahedron Ca2+ model

The intracellular Ca2+ release mechanism functions, as introduced in section 2.1,by subsequent openings of IP3 receptor clusters. These IP3Rs are sensitive bothtowards IP3 , which concentration is controlled by an external stimulus, and Ca2+

, which concentration is dependent on already open IP3Rs . This calcium inducedcalcium release (CICR) mechanism constitutes a positive feedback, where theopening of a single IP3R may eventually trigger a global Ca2+ wave. Such anevent is characterized by the short-lived cellular state where the majority of theIP3R clusters are open, evoking a sharp Ca2+ concentration spike. The CICRmechanism heavily relies on spatial coupling between the IP3Rs , mediatedby Ca2+ diffusion. The diffusion profile upon the opening of an IP3R hadbeen characterized as very steep [101]. This effectively means, that the IP3Rsconstituting a cluster are almost certainly coupled, whereas all other more distantIP3R clusters only open with a certain probability determined by the local Ca2+

concentration. This opening probability is therefore generally dependent on thenumber of already opened IP3R clusters and their respective distances. Themodel under consideration here exploits the spatial symmetry of a tetrahedralarrangement and consequentially consists of only four clusters [103]. All stateswith the same number of open clusters are equivalent with respect to theresulting Ca2+ concentration profile. For this specific setting, the clustersare interchangeable and the opening probabilities are only dependent on thetotal number of open clusters as they are all equidistant. In summary thesystem consists of N = 5 distinct states S0, ..., S4 ordered in a linear chain,corresponding to (0, 1, 2, 3, 4) open clusters. A schematic overview of the modelis shown in figure 2.4. The HSM approach applies to the cluster opening andclosing probabilities, the ’puffs’, which are described by non-exponential waitingtimes. The microscopic single channel dynamics, the ’blips’, within a clusterare not explicitly considered, which greatly reduces the state space. A Ca2+

spike is defined to occur when the system is in state S4. The ISI is thereforedefined by the time the system takes for the transition from state S0 to state S4

69


and corresponds to the FPT F0,4. The moments of the ISI distribution can beanalytically calculated according to the method devised in the previous section2.2.3. Before constructing the model, the meaning of such an idealized and smallCa2+ model shall be discussed briefly. Firstly, the initiation of a global cellularCa2+ release wave, called wave nucleation, often arises from an initial event witha few clusters involved. Preferred nucleation areas have indeed been observed,e.g. in hepatocytes [25]. The minimal number of open clusters causing a globalspike with almost certainty is called the critical nucleus. In that sense, modelingonly a small number of clusters can be sufficient to capture the global Ca2+

dynamics. Secondly, this simplified Ca2+ model has led to powerful predictionswhich were experimentally verified [105] and which shall be discussed later inthis work in section 2.5.

Figure 2.4.: Schematic state space and transition overview of the tetrahedronCa2+ model, constituting a linear chain. Four clusters are considered,which due to symmetry make up to five states, corresponding to(0, 1, 2, 3, 4) open clusters, in total. The interchangeable systemstates are depicted as sets of four circles each, a • marks a closedcluster and a ◦ marks an open cluster. Without the enforcement ofspatial symmetry, the system would consist of up to 16 disparatestates.

At first, the waiting time density for the closing of a single IP3R cluster,Ψc(τ) shall be examined. As stated earlier, such a cluster consists of a numberof single IP3Rs acting as Ca2+ channels. The individual channels in a clusterclose independently with closing rate γ, which does not depend on the Ca2+

concentration and is a molecular property of the IP3Rs [91]. The channel closingrate has recently been determined by total internal reflection fluorescence (TIRF)microscopy and is γ = 59 s−1 in SH-SY5Y cells [91]. Based on that, the waitingtime density for closing of an IP3R cluster with on average Nch channels involved

70


in a puff is given by:

Ψc(τ) = Nchγe−γτ (1 − e−γτ )Nch−1. (2.28)

For the opening probability densities, no such straightforward derivation asfor the closing times is possible. Their general form has been experimentallyassessed by inter puff interval analysis of single IP3R clusters [104], and ap-proximations by gamma distributions have shown reasonable simulation results[103]. Here another two-parameter distribution, termed generalized exponential(GE) function [42], shall be used to capture the opening transition of a singlecluster Ψo(τ). The main reason for this specific choice is, that GE functions arebetter suited for an analytical treatment involving Laplace transformations, asneeded to solve for the FPT problem. Besides, their closeness to the gammadistribution has been established [41]. It reads

Ψo,i(τ) = αiλi(1 − e−λiτ

)αi−1e−λiτ , (2.29)

α ≥ 1 is the shape parameter and λ > 0 is the scale parameter. The waitingtime distributions depend on [Ca2+] and consequently on the number of openclusters. Therefore a subscript i is added, i.e. Ψo,i which describes the openingprobability of a cluster at system state i. The determination of the dependenceof the parameters αi and λi on cellular parameters is based on computations bya method developed in an earlier study by the Falcke lab [46, 102]. The methoduses the De Young-Keizer model [22] for the description of the individual IP3Rs .On the basis of that model, the Ψo,i’s can be computed from the master equationdescribing the random channel state changes. Briefly, the De Young-Keizermodel assumes that a channel is open when three out of the four subunits ofthe IP3R are bound by IP3 and activating Ca2+, but not by inhibiting Ca2+.The transition rates between the states could be determined by experimentsto some extent. Analytical approximations to the simulation results then givefunctional relations of the form αi(x) and λi(x), where x = [Ca2+], [IP3], Nch.Details can be found in the SI of ref. [72]. The opening of the very first cluster,a puff, is well described by a Poisson process which has a exponential waitingtime density [103, 104]. Hence, for Ψo,0 one has α0 = 1 and the puff rate wasexperimentally [104] determined to be λ0 = 0.31s.

The probability not to have left a state by time τ after arrival at time 0 isψo,c(τ) = 1−∫ τ

0 ψo,c(τ ′)dτ ′, for opening or closing of a single cluster, respectively.It is noteworthy to restate, that in accordance with the frame work of semi-Markov processes introduced in section 2.2.2, the time variable τ does notcorrespond to a systems time, but describes the wait after the last transition tostate i. A specific transition from i to i ± 1 open clusters, is the product of theprobabilities that all but one cluster remain at their respective states multipliedby a opening or closing waiting time density:

Ψi,i+1 = (4 − i)ψo,i ×(ψc

)i ×(ψo,i

)3−i

Ψi,i−1 = iψc ×(ψc

)i−1 ×(ψo,i

)4−i, (2.30)

71


where the prefactors account for the multiplicity of the transitions. The individ-ual opening time and closing time probability densities as well as the constructedconditioned waiting times Ψi,j are shown in figure 2.5.

0

2

4

6

8

0 0.06 0.12

dens

ity(s

-1)

time (s)

B

0

12.5

25

0 0.25 0.5

dens

ity(s

-1)

time (s)

AA

Figure 2.5.: Individual opening time and closing time probability densities(Ψo,i, Ψc) and the conditioned waiting time densities for the con-secutive opening transitions, constructed out of the closing (Ψc)and opening (Ψo,i) densities for the tetrahedron model according toequation 2.30. The colors in the legend depict the system state interms of the number of open clusters (Si, i = 0, 1, 2, 3, 4).

Now that the transition network for the Ca2+ model, constituting a simplelinear chain, has been fixed, the solution strategy for the FPT problem can beexplicitly employed. At first the set of probability fluxes shall be calculated. Bywriting the set of fluxes as a vector Ij(t) and by writing the conditioned waitingtime densities in an appropriate matrix Ψ(t), the system of integral equationsgiven by equation 2.9 can be written in Laplace space as

(� − Ψ)Ij = f i. (2.31)

This is an inhomogeneous linear system of equations for the Laplace trans-formed fluxes Iil which can be solved exactly by standard algebraic meth-ods. The solution yields all Laplace transformed fluxes as functions of theLaplace transformed waiting times. The flux vector can be written as Ij =(Ij

01, Ij10, Ij

12, ..., IjN−1,N , Ij

N,N−1) with N = 4 and the initial function vectors forthe r.h.s. of Eq. 2.31 are f0 =

{δi0Ψ0,1

}and f8 =

{δi8Ψ4,3

}, i = 1, . . . , 8. The

waiting time matrix Ψ is completely determined by the set of equations givenin Eq. 2.9 and reads

Ψ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 Ψ0,1 0 0 0 0 0 0Ψ1,0 0 0 Ψ1,0 0 0 0 0Ψ1,2 0 0 Ψ1,2 0 0 0 0

0 0 Ψ2,1 0 0 Ψ2,1 0 00 0 Ψ2,3 0 0 Ψ2,3 0 00 0 0 0 Ψ3,2 0 0 Ψ3,20 0 0 0 Ψ3,4 0 0 Ψ3,40 0 0 0 0 0 Ψ4,3 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (2.32)

72


As can be seen in Eq. 2.31 there are actually two sets of fluxes, reflecting thetwo distinct initial states needed for the FPT calculation based on the renewalequation. This formula 2.14 simplifies for the tetrahedron model to

F0,4 =I0

3,4 − I04,3

I43,4 − I4

4,3 + 1. (2.33)

Plugging in the solutions of the linear system given in equation 2.31 yields theLaplace transformed FPT density:

F0,4 =Ψ0,1Ψ1,2Ψ2,3Ψ3,4

1 − Ψ0,1Ψ1,0 − Ψ1,2Ψ2,1 − Ψ2,3Ψ3,2 + Ψ0,1Ψ1,0Ψ2,3Ψ3,2(2.34)

It is an algebraic expression consisting solely of Laplace transformed transitionwaiting times Ψi,j(s). The Laplace transformations of the underlying openingand closing waiting times are given in the Appendix B.1. The first momentof this FPT density corresponds the average ISI (Tav ) in the tetrahedronCa2+ model. In practice, the analytical approximations for the waiting timedistribution parameters [72] are used to explicitly set up the Ψi,j ’s depending oncellular parameters like [IP3] or the average number of channels in a cluster Nch.Then, assisted by a computer algebra program [113], Tav can be calculated usingequation 2.16 of the previous section. Results are shown in figure 2.6. Equation2.16 also allows to analytically calculate higher moments of the ISI distribution.By calculating the 2nd moment < t2 > it is possible to compute the Coefficientof Variation (CV), given by Tav/σ with σ denoting the standard deviation.Additionally, by calculating the 3rd moment < t3 > the skewness, given by< t3 > /σ3, of the ISI distribution can be computed as well. An examination ofboth quantities, as shown in figure 2.7, identifies the ISI distribution to be ofexponential type for the parameters used in the tetrahedron model. This is inagreement with the earlier result of ref. [103] that a global feedback is necessaryto reach the regime σ < Tav , which is typically observed in experiments [90].Unfortunately, global feedback does not allow for an exact analytical solution forthe moments of the FPT density, as it makes an explicit systems time necessary.In other words, the ISI distribution is a rescaled puff distribution given by Ψo,0,which is exponential. The reason for that is the time scale separation betweenthe slow single puff dynamics and the very fast consecutive cluster openingdynamics. As indicated in figure 2.5, the mean waiting time between single puffs,determined by the conditioned waiting time Ψ01, is of the order of 1/4λ0 ∼ 1 s.Whereas the mean waiting time for the consecutive cluster openings, determinedby Ψ12, Ψ23 and Ψ34, is of the order of ∼ 0.05 s. By artificial lowering thechannel closing rate γ, this time scale separation becomes weaker. As depictedin figure 2.7 the ISI distribution then clearly deviates from the exponentialdistribution, and this effect gets even stronger with growing puff rate λ0. Thisobservations will be deepened and used to construct a much simpler Ca2+ model,called generic model, in the next section.

73


0

100

200

300

400

0.5 0.75 1 1.25 1.5

4 5 6 7 8 9 10T a

v(s

)

IP3 (μM)

NchA

Tav(IP3)Tav(Nch)

Figure 2.6.: The average ISI (Tav ) for the tetrahedron Ca2+ model in dependenceof two cellular parameters. Different [IP3] correspond to differentexternal stimulation strengths, the stronger the stimulus the lowerthe Tav and therefore the faster the spiking. Nch is the averagenumber of channels a IP3R consists of, more channels lead to highertotal opening probabilities and therefore also to shorter averageISIs. Tav is analytically calculated as the first moment of the firstpassage time density defined in the main text.

2.3. Exploiting time scale separation - The generic Ca2+

model

As discussed in the previous section, the single puff dynamics constituting theS0 → S1 transitions are about 20 times slower, than the consecutive clusteropenings described by the Si → Sj transitions with i, j = (1, 2, 3, 4). These arevery fast and may potentially lead to a Ca2+ spike with all clusters open. Thismeans, that the system upon being in state S1, with respect to the puff timescale, almost instantaneously decides with a certain probability if this singlepuff becomes a global spike, or if it relaxes back to the ground state S0 withoutreaching S4. This probability is denoted the splitting probability C14 and isgiven by [103]

C14 =C12C23C34

1 + C12(C23 − 1) + C23(C34 − 1), (2.35)

with the one-step splitting probabilities Ci,i+1 =∫∞

0 Ψi,i+1(t)dt. This expressionis derived by considering all possible transition routes from S1 to S4 withouttouching the ground state S0. This probability can be calculated in dependence

74

2.3. Exploiting time scale separation - The generic Ca2+ model

1.4

1.6

1.8

2

0 20 40 60

skew

ness

γ (s-1)

B

0.4

0.6

0.8

1

0 20 40 60

CV

γ (s-1)

C

Figure 2.7.: Skewness and Coefficient of Variation of the ISI distribution ofthe tetrahedron Ca2+ model. The parameter values used for thestochastic Ca2+ model are indicated by the arrows. For these,the ISI distribution is of exponential type, as the CV indicatesσ = Tav and the skewness is exactly two. By artificially varyingthe channel closing rate γ the time scale separation between singlepuff dynamics and consecutive openings gets weaker, and the ISIdistribution clearly deviates from an exponential distribution. Thiseffect grows stronger with increasing puff rate λ0.

on cellular parameters with the setup of the tetrahedron model, results regardingthe IP3 concentration and the number of channels Nch are shown in figure 2.8.Pursuing this perspective on the system lead to the idea to separate the stochasticCa2+ spike generating process into one Poisson process describing the puffs,and a Bernoulli trial with success probability C14. This effectively constitutes asplitting of an inhomogeneous Poisson process which shall be elucidated in thefollowing.

2.3.1. Model Construction

At first, the puff process shall be modeled more realistically by incorporating anegative feedback. The now time dependent puff rate is recast as

λ(t) = λ0(1 − e−ξt). (2.36)

The feedback ensures that the probability for a puff to occur at t = 0 is equal tozero. This better reflects physiological constrictions like internal Ca2+ storagedepletion which inhibit puffs right after a global Ca2+ spike. The probabilityfor a puff then recovers with a rate given by ξ. Equation 2.36 constitutes aninhomogeneous Poisson process for the Ca2+ puff dynamics. The specific formof the feedback is borrowed from ref. [90], where it was used to incorporate aglobal negative feedback on the global Ca2+ spike rate.

One can now ask for the probability that at a time point t a puff occurs andtriggers a global Ca2+ spike. Conditioned on the ground state S0 the time for

75


0

0.01

0.02

0.1 1 2 3 4 5 0

0.1

0.2

0.3 1 5 10 15 20

C14

C14

IP3 (μM)

Nch

C14(IP3)C14(Nch)

Figure 2.8.: Parameter dependencies for the splitting probability C14 to reachthe spiking state S4 out of a single puff state S1 for the tetrahe-dron model. The axis are differently scaled for the cytosolic IP3concentration ([IP3] ) and the mean number of Ca2+ channels percluster participating in an opening event (Nch) respectively. Thesplitting probability saturates for both parameters, however for theNch dependency only for a unrealistic high number of Ca2+ channels(Nch >100)

the first Ca2+ spike to occur, the ISI, is of interest. Formally this leads to:

ps(t) = C14λ(t) ×(e−Λ(t)

+(1 − C14)∫ t

0e−Λ(t1)λ(t1)

+(1 − C14)2∫ t

0

∫ t

t1e−Λ(t1)λ(t1)e−(Λ(t2)−Λ(t1))λ(t2)e−(Λ(t)−Λ(t2))dt1dt2

+ ...)

= C14λ(t)e−Λ(t)(1 + (1 − C14)

∫ t

0λ(t1)dt1

+ (1 − C14)2∫ t

0

∫ t

t1λ(t1)λ(t21)dt1dt2 + ...

)(2.37)

This expression contains the probabilities for all possible puffs occurring before tnot leading to a spike, the failed Bernoulli trials with probability (1 − C14). ThePoisson intensity at a time t is given by Λ(t) =

∫ t0 λ(t). The very first term is

the probability that at t a puff occurs and becomes a global spike and no otherpuff occurs in [0, t), the second term is the probability for one failed puff in [0, t),

76


the third term handles two failed puffs and so on. Following permutation andsymmetry arguments outlined in the book by Van Kampen [107] one may write:

ps(t) = C14λ(t)e−Λ(t)( ∞∑

n=0

1n!((1 − C14)Λ(t)

)n)

= κ(t)e−C14Λ(t). (2.38)

The resulting stochastic process for the Ca2+ ISIs is also an inhomogeneousPoisson process with the spike rate κ(t) = C14λ(t) and intensity Λ(t) =

∫ t0 λ(τ)dτ .

The new rate is just the puff rate refined with the splitting probability C14,this property is known as Poisson splitting. By writing the global spike rate asκ(t) = C144λ0(1 − eξt) it is clear, that this approach yields an expression forthe global Ca2+ spike rate: κ0 = C144λ0. The specific parameterization andstructure of the tetrahedron model is by no means necessary for the genericCa2+ model and its structure can be in principle generalized to multiple puffsites with individual splitting probabilities:

κ0 =N∑

i=1λ0,iC1N,i. (2.39)

This equation combines the local puff dynamics, determined by the λ0,i’s, andthe spatial coupling of the cluster arrangement, given by the C1N,i’s, to yield asimple expression for the global Ca2+ spiking rate. Accompanied by the feedbackthe generic model gives results which are in good agreement with experimentaldata, as to be seen in the following.

2.3.2. Results of the generic model

The great advantage of the generic model is its capability of providing a closedexpression for the ISI distribution, incorporating a global negative feedback andthe local puff dynamics. This allows for analytically reproducing the momentrelation between the standard deviation σ and the average Tav of spike trainsfound experimentally.

As the algebraic structure of the ISI distribution of the generic model, termedps(t) in the preceding subsection, is the same as in ref. [90], the analyticalexpressions for the moments of the ISI derived in that work can be readilyapplied here. They read for the tetrahedron model setup:

⟨t1⟩ = Tav =

eΘ(Θ)1−Θ

4λ0C14

[Γ(Θ) − Γ

(Θ, Θ

)],

⟨t2⟩ =

2eΘ

(4λ0C14)2 F[(

Θ, Θ),(1 + Θ, 1 + Θ

), −Θ

], (2.40)

where Θ = 4λ0C14/ξ, Γ(x) denotes the Euler gamma function, Γ(x, y) theincomplete gamma function and F [x] is the generalized hypergeometric function.

These equations allow for an analytical analysis of the moment relationbetween Tav and σ for different feedback strengths ξ. As shown in figure 2.9,

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0

100

200

300

400

500

0 100 200 300 400 500

σ(s

)

Tav (s)

ξ=0.001 s-1

ξ= 0.01 s-1

ξ= ∞ s-1

Figure 2.9.: The σ-Tav relations calculated with the generic Ca2+ model fordifferent feedback strengths. The relations are linear with the slopebeing cell type specific and controlled by ξ. The position of anindividual cell on a moment relation is determined by cell specificproperties like cluster arrangement. This cell-to-cell variability iscaptured by the splitting probability C14 which compactly accountsfor all factors contributing to the spatial coupling of the channelclusters. Individual values of Tav for three different values of C14 (•0.05, � 0.01, � 0.008) were added to each line. Modifications of thecoupling strength merely lead to a shift of the cell along the σ-Tavrelation and do not affect the slope of the relation.

this relation is linear with the slope controlled by ξ, which is considered as celltype specific [90, 103]. This is exactly what is observed experimentally when acell population is analysed, and the σ-Tav relation is calculated for every singlecell. Interestingly, the theoretical information content of Ca2+ spike trains issolely determined by the slope of this σ-Tav relation, which in turn is a robustproperty of a cell population [89].The individual position of a cell on the momentrelation is determined by cell specific properties, like cluster arrangement. Thiscell-to-cell variability is in the generic model completely captured by the splittingprobability C14 which condenses all factors contributing to the spatial couplingof clusters into one quantity. Variation of the coupling strength mediated by thevalue of C14 leads to a shift of the cell along the σ-Tav relation and does notaffect the slope of the relation. This behavior was found in buffer experiments[89], where different buffer concentrations decreased the spatial coupling of theIP3R clusters. This affected both σ and Tav of the recorded spike trains and

78


shifted individual cells in the σ-Tav plane with a slope similar to the slope ofthe moment relation obtained from a population of unperturbed cells.

The generic model is only an approximation of the real stochastic processgoverning the Ca2+ spike generation. Although, as demonstrated, the resultsare in good agreement with experiments, a quantification of the error of theapproximation would be eligible. This shall be done in the following subsection.

2.3.3. Error analysisThe applied Poisson splitting ansatz completely neglects the dynamics of thesystem between the individual states, instead it implicitly sets all transition timesexcept for the S0 → S1 transition to zero. So it is expected to underestimate thetrue average ISI, T gm

av < T trav, with the generic Ca2+ model. The time the system

spends in processing the failed puffs, the ones which do not reach the higheststate S4, is a major source for the underestimation. The number of those failedpuffs is given on average by 1/C14, as these are the odds for the Bernoulli trialexperiment to fail. For standard parameters this accounts for ∼ 100 failed puffsto occur on average before a successful spike occurs. So the error made by usingthe Poisson splitting is expected to scale exactly with 1/C14. By letting therecovery rate approach infinity, ξ → ∞, the negative feedback is turned off, andthe results of the generic model are comparable to the exact analytic resultsobtained with the presented FPT formalism for the semi-Markovian tetrahedronmodel of section 2.2. As shown in figure 2.10 the absolute error of the averageISI, given by T tr

av − T gmav , indeed scales linearly with 1/C14 and is of the order

of ∼ 10 seconds. The relative error in dependence on the IP3 concentration,given by (T tr

av − T gmav )/T tr

av, was also computed and remains below 10% for thephysiological relevant parameter range. Additionally, the relative error saturatesfor IP3 → ∞ and approaches zero for IP3 → 0. The main argument for thegeneric model is the time scale separation between single puff dynamics and theconsecutive cluster openings. In accordance with that the approximation errorgets bigger with weaker time scale separation, realized by a smaller channelclosing rates γ as indicated in the figures 2.10.

In summary the main source for the approximation error was identified as theneglected cluster state dynamics after a puff not leading to a spike. This errorwas quantified by turning off the feedback in the generic model and comparingthe results for Tav to the exact analytic results for the tetrahedron model. It issmall enough to make the Poisson splitting ansatz valid for describing the Ca2+

spiking dynamics, as long as the time scale separation between puff and spikedynamics is strong enough. It gets even smaller when compared to numericalresults for the tetrahedron model with global feedback, as presented in thefollowing.

79


0

3

6

9

12

15

70 120 170 220

erro

r(s)

1/C14

γ = 58.8 s-1

γ = 50 s-1

γ = 40 s-1

0.02

0.08

0.14

0.5 2 3.5 5

rela

tive

erro

r

IP3 (μM)

B

γ = 58.8 s-1

γ = 50 s-1

γ = 40 s-1

Figure 2.10.: Absolute (right) and relative (left) errors in Tav of the genericCa2+ model, obtained by comparison with the exact results of thesemi-Markovian tetrahedron model. The number of failed puffs ison average 1/C14 and the absolute error scales accordingly, as thesedynamics are completely neglected by the Poisson splitting ansatzof the generic model. The relative error remains below 10% forthe physiological relevant parameters and is bounded. Weakeningthe time scale separation by artificially varying the channel closingrate γ increases the error.

80

2.4. Numerical Analysis of the HSM Ca2+ model


A serious limitation of the analytical treatment of the tetrahedron model insection 2.2 is the impossibility of incorporating a global feedback. This is dueto the formal constraints of the semi-Markovian framework: writing Ψij(τ)for a conditioned waiting time the variable τ describes the time it will taketo leave state i and land at state j. The instant τ = 0 describes the eventof the system just arriving at state i, there is no systems time which couldfor example monitor how long it has been since the spiking state S4 was lastvisited. However, exactly this information is needed to incorporate a globalfeedback which introduces a process of recovery after a Ca2+ spike has occurred.As approved by analytical calculations in section 2.2 and the results of thegeneric model of the preceding section, such a feedback is required to reproducethe non-Poissonian σ-Tav relations found experimentally. As demonstrated infigure 2.7, non exponential ISI distributions can be obtained in principle withinthe semi-Markovian tetrahedron setup. However, the exponential puff waitingtime Ψ01 in combination with the strong time scale separation enforced by thephysiological parameters do not allow for that. However, as done in ref. [103],by falling back to numerical procedures to compute the ISIs a systems timeand therefore a global feedback can be readily introduced into the tetrahedronmodel. A standard method for the simulation of Markovian systems is theGillespie algorithm [36], as to be seen in the following with a slight enhancementis also applicable for semi-Markovian systems. At first, an already establishedalgorithm shall be applied and the results compared to the generic model withglobal feedback.

2.4.1. The DSSA algorithm

An algorithm to simulate the semi-Markovian tetrahedron system was termedDelayed Stochastic Simulation Algorithm (DSSA) and developed in ref. [103]. Itreadily simulates realisations of the Ca2+ spiking process, an example is shownin figure 2.11. The global feedback is introduced in the puff transition waitingtime by explicitly memorizing the time of the last spike, stored in tsp, and itadditionally also keeps track of the last state transition time, stored in tp. Theopening waiting time density for the first cluster to open is therefore given by[103]:

Ψo,0(t, tp, tsp) = λ0(1 − e−ξ(t−tsp)) e

−λ0∫ tsp

tp1−e−ξ(τ−tsp)dτ

(2.41)

= λ0(1 − e−ξ(t−tsp)) e

− λ0ξ

[−e−ξ(t−tsp)+e−ξ(tp−tsp)−ξ(t−tsp)]

Here, the variable t corresponds to the total systems time and the algorithmworks by iterating over all timesteps between t = 0 and a total time T witha fixed step size dt. For every step of the iteration and for every clustera random number Uj is drawn from a uniform distribution U(0, 1) and therespective transition probabilities for every cluster j are calculated according toPj(t) =

∫ t−τ+dtt−τ Ψo/c,j(t′)dt′. The transition type is determined by the actual

81


state of cluster j, so Ψo if the cluster is closed and vice versa. If Pj(t) > Uj forsome epoch t, the cluster state is changed accordingly and the the time of thelast transition is updated to τ = t. A more detailed description of the algorithmcan be found in the SI of ref. [103].

0

1

2

3

4

0 200 400 600 800 1000

Ncl

time (s)

D

Figure 2.11.: Stochastic simulations of spike trains of the tetrahedron with theDSSA algorithm for [IP3] = 1 μM. There are frequent puff eventswith less than four clusters open in contrast to the rather isolatedglobal spike events characterized by Ncl = 4. Such simulationswith runtimes up to 107s can be used to compute ISI statistics bymeans of sample mean and sample variance.

By running very long Monte Carlo simulations, values of Tav and σ canbe reliably computed by means of the sample mean and sample variance independence on cellular parameters. Their goodness has been approved bycomparing this results to the analytically tractable case already in ref. [103].Here the simulations shall be used, to demonstrate the accuracy of the genericmodel introduced in the preceding section with the global feedback present. Asshown in figure 2.12, its results are very close to the numerical results for Tav. The approximation error gets even smaller for increasing feedback strengths,which cover the biologically more relevant cases as these produce σ-Tav relationsmore close to experimental data as discussed in the preceding section. Thereason for that is simply, that a strong global feedback introduces a longerrecovery after a Ca2+ spike and therefore increases the time scale separationbetween puff and consecutive cluster opening dynamics. This in turn lowersthe relative amount of time the system spends on transitions not covered bythe Poisson splitting ansatz of the generic model. These results confirm theeffectiveness and elegance of the generic model, as it produces results for themoments of the ISI distribution as good as the Monte Carlo simulations of thefull model with muss less effort.

The DSSA algorithm works fine for the specific setup of the tetrahedron Ca2+

82


0

200

400

600

800

0.4 0.6 0.8 1 1.2 1.4 1.6

T av

(s)

IP3 (μM)

ξ = 0.001 s-1

ξ = 0.01 s-1

ξ = ∞ s-1

Generic Model

Figure 2.12.: Comparison of numerical results obtained with the DSSA algorithmfor the IP3 dependence of Tav with the results obtained from thegeneric Ca2+ model with global feedback. The value of ξ introducesa recovery rate of the puff probability after a global Ca2+ spikehas occurred, the smaller ξ the longer the recovery and hence thestronger the global negative feedback. The analytical results ofthe generic model are very close to the simulation results, with theapproximation error getting smaller for stronger feedbacks. Forξ =0.001 the values for Tav are almost identical, confirming thegoodness of the generic model also for the biological more relevantcases with enabled feedback.

model. However, it is not readily applicable to general semi-Markovian systemsas it makes no explicit use of the conditioned waiting times Ψi,j . Therefore amore general algorithm for semi-Markovian systems with discrete states shallbe developed in the following.

2.4.2. An exact semi-Markovian simulation algorithm

The great advantage of the Gillespie algorithm (ref. [37] is a good introduction)is, that it is exact for discrete states. The DSSA algorithm is not exact in thatsense, because it introduces a numerical time discretization with a step size dtand is therefore missing all possible transitions with a transition time smallerthan the step size. Additionally, the original Gillespie algorithm needs exactly Niteration steps for a realisation with N transitions, which makes it in combinationwith the cost-effective sampling from an exponential waiting time distributionvery computationally efficient. The DSSA algorithm, on the contrary, iterates

83


over all T/dt timesteps, irrespective of system state changes. So the goal is todevise an algorithm which shares more of the favorable properties of the originalGillespie algorithm and is generally applicable to semi-Markovian systems. Asto be seen in the following this actually only requires minor changes of theoriginal algorithm.

As discussed in section 2.2.2, semi-Markovian processes possess an embeddedMarkov chain with the transition probabilities given by qij =

∫∞0 Ψij(τ)dτ .

This is completely analogue to pure discrete state Markovian systems, andin accordance to the original algorithm the next state of a realisation can becomputed by simply sampling from the discrete distribution given by the setof the transition probabilities {qij} for every state i. What is different is thedetermination of the next jump epoch. For a pure discrete state Markoviansystem, when being in state i this epoch is determined by sampling from theexponential waiting time density, often called sojourn time in the literature,given by Φi(τ) =

∑k,i�=j qik e

−∑

k,i�=jqikτ . In the semi-Markovian case (see also

section 2.2.4), having already determined the next state j, the next jump epochis given by sampling from the distribution Φi,j = 1

qijΨij(τ), which is generally

a non-exponential probability density function. This step will generally bemore costly than in the original Gillespie algorithm, where the inverse samplingmethod can be used. In cases where the inverse of the distribution belongingto the waiting time Φij is not available, other sampling methods like rejectionsampling may be used to generate the next jump epoch. Having determined thenext state and the next jump epoch makes the algorithm complete and it readsin pseudo code:

Define conditioned waiting time densities Ψij(τ)Calculate transition probabilities qij =

∫∞0 Ψij(τ)dτ

Set total simulation time TSet initial system state iSet t = 0while t < T do:

sample next state j from the discrete density given by the {qij}sample next jump epoch τ by sampling from the continuous density Φij(τ)Set t = t + τSet new systems state i = jRecord as needed transition epoch t and state j

The output of this algorithm are the cumulative transition times (t0 = 0, t1 =τ1, t2 = τ1+τ2, ..., tN =

∑N0 τi) and the respective system states (S(t0), S(t1), ..., S(tn)).

For systems with many or even an infinite number of states, the algorithm canbe augmented by subprocedures which generate the Ψij ’s inside the main loopaccording to rules which specify the stochastic process.

This algorithm needs exactly N iterations for a realisation with N transitionsand is also exact in the sense the original Gillespie algorithm is exact as there isno time discretization needed. Given a total simulation time T , the expected

84


number of iteration steps needed is given by the expected number of transitionevents happening in the interval [0, T ]. It is clear, however, that this algorithmgenerally outperforms the DSSA algorithm. For that, the number of iterationsneeded is exactly T/dt, with dt being the step size for the time discretization. Asthe step size dt has to be significantly smaller than the average inter-event time<τev> of the stochastic process to achieve a reasonable numerical precision, theexact semi-Markovian algorithm presented above will on average be <τev>/dttimes faster than the DSSA algorithm. For the parameters used for the MonteCarlo simulations shown in figure 2.12, this speedup is about a factor of 100.Setting up the Ψij ’s for the specific tetrahedron Ca2+ model is a bit tedious,as the conditioned waiting times according to equations 2.30 are products ofpowers of individual density functions for this system. But this has to be doneonly once, and after that the algorithm produces reliable results as shown infigure 2.13.

Figure 2.13.: Validation of the exact semi-Markovian algorithm devised in themain text by comparison to an analytical result obtained by solvingthe FPT problem for the tetrahedron model with [IP3] = 1μM .The sample standard deviations of the algorithmic solutions werecalculated by simulating 50 realisations for every total simulationtime indicated on the x-axis.

Incorporating a global feedback for the puff dynamics can be done akin to theDSSA algorithm by memorizing the time of the last spike tsp and introducing ainhomogeneous Poisson rate by λ(τ, t, tsp) = λ0 (1 − e−ξ(τ+t−tsp)). The systemstime t in this algorithm is only updated when a transition occurs, so that thetotal distance in time from the last spike event will be the actual jump epoch τ

85


plus the systems time t, holding the last S1 → S0 transition time, minus thetime of the last spike tsp. The waiting time density for the first cluster openingthen reads:

Ψo,0(τ, t, tsp) = λ0(1 − e−ξ(τ+t−tsp)) e−λ0

∫ τ

0 (1−e−ξ(τ+t−tsp)(2.42)

= λ0(1 − e−ξ(τ+t−tsp)) e−λ0τ e

−λ0ξ

e−ξ(t−tsp)(e−ξτ −1).

In summary an exact and cost-effective simulation algorithm was devised byextending the Gillespie algorithm with respect to the determination of the nextjump epoch by non-exponential waiting time densities. It is generally applicableto all well defined semi-Markovian processes with discrete states.

2.5. Encoding Stimulus intensities in random spike trains

Having established a profound understanding about the intracellular Ca2+ spikegenerating process, the next question is how cells actually reliably encodeextracellular stimulus intensities with these stochastic spike sequences. Earlierstudies analyzed the theoretical information content of sequences originating froman inhomogeneous Poisson process [90, 103], and found that this informationcontent is solely dependent on the slope of the σ-Tav relation. This slopeis determined by a refractory global negative feedback as discussed in thepreceding sections, and is a cell type specific robust property. However, giventhe huge cell-to-cell variability found for the characteristics of the individualspike sequences, e.g. for the average ISI, no direct and absolute relation betweenstimulus intensity and Ca2+ spiking could be established on a single cell leveluntil recently. The pioneering idea that a fold change in the average ISI (Tav )reliably encodes stimulus intensity changes was first deduced from theoreticalstudies of the hierarchic Ca2+ spiking model, and then confirmed by extensiveexperimental studies [105]. Accordingly the theoretical considerations shall bepresented first followed by a brief review of the experimental results.

2.5.1. Theoretical predictions

For all theoretical analysis of the intracellular Ca2+ spiking done in this workthe Ca2+ signaling pathway is simplified by only considering processes whichare downstream of the IP3 formation. This means that processes involving thebinding of the extracellular ligand to the G protein coupled cell surface receptorsas well as the subsequent cleavage of membrane phospholipids to IP3 are notexplicitly considered in the model. This includes potentially important factorslike surface receptor density or IP3 activation. The mathematical Ca2+ modeltakes the concentration of active IP3 directly as input, and treats different IP3concentration as different stimulus intensities. However, this simplification canbe expected to be reasonable by making the rather weak assumption that theintracellular IP3 concentration is a monotone function of the extracellular ligandconcentration. An additional simplification applies to the structure of the ISIs.In real cellular Ca2+ spike sequences, the interspike intervals are comprised of

86


the spike duration, a physiological refractory period and the stochastic period.The former lead to a minimal ISI (Tmin). For the mathematical description bya stochastic process, the deterministic Tmin is neglected and only the stochasticperiod is considered. For low to intermediate stimulus intensities, this stochasticperiod constitutes the major component of the ISI [105].

To assess the relative impact of different stimulus intensities on the modeledCa2+ spike sequences, the change of Tav for two different IP3 concentrations,ΔTav = Tav1−Tav2, is analyzed. The values of Tav (IP3 ) are calculated using thegeneric model and, similar to the calculations shown in figure 2.9, the couplingstrength C14 is varied to mimic the cell-to-cell variability. As shown in figure2.14 the relative change of the average ISI is a linear function of Tav1. Thisleads to a simple expression, termed the encoding relation, for the change of theaverage ISI by applying a stimulus step of size ΔIP3:

ΔTav = β(ΔIP3) Tav1, (2.43)

with β being the slope of the relations shown in figure 2.14.

Figure 2.14.: Relation between the change of Tav (ΔTav) and the value of theaverage ISI before the second stimulation (Tav1). Shown are an-alytical results for different stimulation steps with the stimulusintensity for Tav1 being [IP3] = 0.5μM . The functional relationshipis clearly linear, which indicates a fold change response given bythe slope of the depicted curves.

Rewriting equation 2.43 by expanding ΔTav = Tav1 − Tav2 as

β =Tav1 − Tav2

Tav1(2.44)

shows that β can be defined as the fold change of Tav when applying a secondstimulus which determines Tav2. The interpretation is that slower spiking cells ofa population still remain slower after an increase of the stimulation and the sameis true for the faster spiking ones. But the relative change of the average ISI isthe same for all cells and is given by the fold change β. This means, that there

87


is no absolute value of Tav encoding a stimulus intensity, but cells do encodestimulus changes by a constant factor which is independent of the value of Tav1 ofa specific cell. This remarkably resembles Weber’s law from sensory perceptiontheory which originates from observations of the just-noticeable difference ofphysical stimuli for human observers [23].

Model calculations also allow to systematically compute the fold changeβ(ΔIP3) in dependence on the stimulation step size by evaluating the r.h.s. ofequation 2.44 for different values of [IP3] . Results are depicted in figure 2.15and show that the larger the stimulation step the larger the fold change. Thatthe value of β is bounded by 0 < β < 1 is obvious by noting that equation 2.44can be written as Tav2 = Tav1(1 − β). The theory states that the actual courseof the β(ΔIP3) functions is dependent on the value of IP3 for the first Tav . Asto be seen in the following, this is not recovered experimentally.

Figure 2.15.: Functional relationship between the fold change β and the stimu-lation step size ΔIP3. The fold change generally grows with ΔIP3and is bounded by 0 < β < 1. These result were obtained byevaluating the r.h.s. of equation 2.44 for different values of IP3. The needed average ISIs were calculated using the analyticalsolution of the tetrahedron Ca2+ model. Theory predicts, that thevalues of β are also dependent on the IP3 -level for the first Tav1:IPstart

3 .

The main prediction of the theoretical considerations presented here, is thatabsolute values of Tav are not informative on a single cell level about an externalstimulus intensity. However, the relative change of the stimulus intensity, in themodel captured by ΔIP3 , is reliably encoded in the fold change β of the averageISI. This fold change is predicted to be the same for all cells and is thereforerobust against the huge cell-to-cell variability in Tav . In the next section, theexperimental effort and results to successfully confirm this prediction shall bepresented.

88


2.5.2. Experiments supporting the fold change encoding hypothesis

Experimental stimulation of intracellular Ca2+ signaling can be achieved in hu-man embryonic kidney (HEK) 293 cells by exposing the cells to carbachol (CCh).This binds to muscarinic acetylcholine cell surface receptors and activates theCa2+ pathway leading to sustained Ca2+ spike sequences which can be recordedfor up to one hour [105]. To test for the fold change experimentally a pairedstimulation protocol was used, in which the HEK293 cells were stimulated withone concentration of CCh and afterwards were exposed to a higher concentrationof CCh by switching the medium. This difference in the CCh concentration(Δ[CCh]) constitutes the stimulus step size experimentally. ISIs were recordedaccordingly and the respective averages Tav1 and Tav2 were individually calcu-lated for all cells. Plotting the results in the ΔTav - Tav1 plane consistentlyallowed for a linear fit with the encoding relation defined in equation 2.43 and βcould be determined for different stimulation step sizes. In the results shown infigure 2.16 two different Δ[CCh] are shown, and the fold change β given by theslope indeed increases for larger steps in the CCh concentration as predictedfrom theory in the preceding subsection, i.e. represented in figure 2.14.

Figure 2.16.: Relation between the change of Tav (ΔTav) and the value of theaverage ISI of the first stimulation (Tav1) for two paired stimulationexperiments [105]. Shown are each two different stimulation stepswith the indicated CCh concentrations in μM . The single dotsrepresent the average ISI values for the spike trains of the singlecells. A linear model according to the encoding relation ΔTav =βTav1 was fitted to the data. The larger step sizes of 170 and 150μM Cch respectively in blue lead to larger slopes compared tothe smaller Δ[CCh] of 120 and 100 μM respectively shown in red.This confirms that the fold change β reliably encodes changes ofstimulus intensities.

These results were further assured by repeating these experiments with anothercell type. Sequentially stimulating hepatocytes with phenylephrine showed asimilar linear relationship in the ΔTav - Tav1 plane. The goodness of the linearmodel expressing the fold changes was assessed by using Pearson’s correlationcoefficient and analysis of explained uncertainty. As shown in figure 2.17 these

89


analyses additionally confirmed the fold change hypothesis.

Figure 2.17.: Pearson’s correlation coefficients (ρ) and explained uncertainties(uex) for the linear model ΔTav = βTav1 constituting the foldchange response. The stimulation steps in CCh concentration (μM)of the HEK293 cells are indicated on the right. The hepatocyteswere stimulated with 0.6 μM and then 1.0 μM phenylephrine. Asa value of 1 for both ρ and uex indicates a perfect linear correlationthe fold change hypothesis is well confirmed for all but the smallestΔ[CCh].

These paired stimulation experiments with HEK293 cells lead to the obser-vation, that the fold change β only depends on the step size Δ[CCh] and notthe initial CCh concentration. This is shown in figure 2.18, where the measuredvalues of β do not support for a dependence of the fold change on the initialCCh concentration. Writing the encoding relation equation 2.43 as a differenceequation

Tav([CCh] + Δ[CCh]) − Tav([CCh]) = −β(Δ[CCh])Tav([CCh]), (2.45)

and making a first order approximation of the fold change dependence on thestimulation step size according to

β(Δ[CCh]) =∂β

∂Δ[CCh]Δ[CCh], (2.46)

finally leads by taking the limit Δ[CCh] → 0 to the differential equation

dTav

d[CCh]= −γ Tav, γ =

∂β

∂Δ[CCh]

∣∣∣∣∣Δ[CCh]=0

. (2.47)

The solution of this equation yields an exponential dependence of the averageISI on the CCh concentration:

Tav = e−γ([CCh] - [CCh]ref ) Tav,ref . (2.48)

90


Figure 2.18.: Measured values of the fold change β of HEK293 cells for pairedstimulation experiments with stimulation step sizes (Δ[CCh]) asindicated on the x-axis. The reference concentrations were [CCh]ref

= 30 μM (red) and [CCh]ref = 50 (blue). The line shows theexponential relationship between the fold change and Δ[CCh]formulated in equation 2.49 in the main text.

The average ISI measured at a reference CCh concentration is given by Tav,ref ,which also captures the differences between individual cells in the response tothe reference stimulus [CCh]ref . The single parameter γ describes the sensitivityof Tav on the stimulus intensity. Reinserting this expression into the encodingrelation shows that β obeys an exponential dependence on the stimulation stepΔ[CCh] = [CCh] - [CCh]ref :

β = 1 − e−γΔ[CCh]. (2.49)

This relation can be reliably fitted to measured values of the fold change β asshown in figure 2.18. However, according to results from the mathematical Ca2+

model, depicted in figure 2.15, there is a dependence of the fold change on thereference CCh concentration. There is currently no straightforward explanationfor this discrepancy. One source of it might be that the model neglects allprocesses of the Ca2+ pathway upstream to IP3 and directly takes the IP3concentration as stimulus input.

By noting that γ according to the fold change hypothesis has to be the samefor all cells in a population, equation 2.48 can be recast on the population levelas

Tpop = e−γ([CCh] - [CCh]ref ) Tpop,ref . (2.50)

Here Tpop and Tpop,ref are the population averages of Tav and Tav,ref respectively.The universality of the exponential concentration-response of Tpop could bestrikingly verified not only by the measured data used for the derivation, but

91


also for published data for hepatocytes [81] and insect salivary glands [76]. Ineach case shown in figure 2.19 the effects of the stimulus intensity on Tpop werewell described by equation 2.50.

Figure 2.19.: Applying the exponential stimulus concentration-response derivedin the main text to different measured data sets. The HEK293data was actually used for the derivation, the other data setshave long been published. Hepatocytes were stimulated witheither phenylephrine or vasopressin [81], the population averagesof the average ISIs (Tpop= <Tav >) nicely follow the exponentialrelationship on the ligand concentration. The same is true for datafrom insect salivary glands stimulated with 5-HT [76].

The experimental evidence gathered to support the fold change hypothesis iscompelling. It showed that there is a constant fraction β by what individualcells change their Tav according to a stimulus step size, and this measuredrelation between Tav1 and Tav2 is well described by the encoding relation firstdeduced from theory. Although the dependence of β on the initial stimulusintensity was not recovered experimentally, this observation in turn lead to aphenomenological derivation of the concentration-response which strikingly anduniversally explains the average ISI for different cell types and stimuli.

92

2.6. Discussion of the stochastic Ca2+ modeling

2.6. Discussion of the stochastic Ca2+ modelingThe initial goal of the investigations presented here was to progress the effortsmade by the Falcke group to adequately model intracellular Ca2+ signaling.Most notably, this included the extension and deepening of the mathematicalconcepts employed in ref. [103]. The established theoretical and experimentalfindings of the Falcke group revealed the immanent stochasticity of the Ca2+

spike trains and identified the ISI distribution as the key descriptor of the singlecell Ca2+ signals [90]. Exploiting spatial symmetries, stationary Ca2+ diffusionprofiles ([101]) and results from fluorescence microscopy of single puff sites ([104])lead to the stochastic tetrahedron Ca2+ model described in ref. [103].

From this perspective, which is the starting point for the main analyticalresult derived in this work, the interspike interval is given by the first passagetime to go from the state of all Ca2+ release sites (the IP3R clusters ) closed tothe state where all release sites are open. To incorporate information from singleCa2+ channel dynamics into the full model but to avoid a state space explosionthe concept of hierarchic stochastic modeling was introduced already in ref.[103]. It relies on partitioning the huge microscopic state space into functionalor observable states. This is nothing new, but by employing non-exponentialwaiting times for the transitions between these functional states as opposed toassigning simple rates, not all information from the microscopic dynamics islost. In section 2.2.2 this concept was embedded in the mathematical theory ofsemi-Markov processes where the conditioned waiting times Ψij were rigorouslydefined. The correspondent Master equation is comprised of a probability fluxbalance, similar to the discretized continuity equation in physics, and the fluxesare given by a recursive integral equation containing the non-exponential waitingtimes [79]. Combining this non-Markovian Master equation with the renewalapproach ([107]) lead to a convenient analytical solution in Laplace space of thefirst passage time problem (section 2.2.3). The possibility to compute arbitrarymoments of the FPT distribution without the need for the often cumbersomeback-transformation from Laplace space accounts for the high practical value ofthis solution. Its application to the tetrahedron Ca2+ model was straightforwardand analytical results for the first four moments of the FPT distribution independence on cellular parameters were calculated in section 2.2.5. The valuesof the moments, in particular the σ ≈ Tav relation, indicate an exponential ISIdistribution for physiological parameter values.

This result was, taken into account the complex semi-Markovian modelingframework, a bit puzzling. However, it could be eventually explained by notingthe time scale separation between the isolated puff events, described by theexponential waiting time Ψ01, and the subsequent cluster openings described bynon-exponential waiting times. Loosely speaking, the puff events get filteredby the very fast subsequent cluster dynamics, and the average fraction of puffssuccessfully culminating to a spike without touching the ground state is givenby the splitting probability C14. This is exactly the Poisson splitting ansatzemployed for the generic Ca2+ model of section 2.3. Poisson splitting states thatgiven a Poisson process with some rate λ(t) and its arrivals are accepted with aprobability p the new process is also a Poisson process with rate pλ(t). From

93


that perspective it is quite evident that an exponential puff waiting time, withconstant λ0, will result in an exponential ISI distribution when splitted withprobability C14. Hence, incorporating a global negative feedback on the puffrate, assuring that λ0(t) = 0, will result in a non-exponential ISI distribution.These are found experimentally and are characterized by a σ-Tav relation witha slope smaller than one [90]. In the model, the feedback strength ξ is given bythe recovery rate of the puff rate. However, the detailed molecular mechanismconstituting this predicted global feedback still remains elusive. In summary,the exploitation of the time-scale separation both explains the performance ofthetetrahedron model without feedback and allows for the generic model, whichanalytically reproduces the experimental findings with much less effort. Thecrucial splitting probability C14 couples the local puff dynamics to the spatialcluster arrangement which in combination determine the Ca2+ spiking process.It also conveniently allows to introduce cell-to-cell variability, e.g. of clustercoupling strength, into the model. For this work, the dependence of C14 oncellular parameters like stimulus strength was adopted from the tetrahedronmodel. Other parameterizations or inhomogeneous puff sites, i.e. multiple λi,o’s,offer plausible extensions of this general approach. The validity of the genericmodel was confirmed analytically (section 2.3.3) and numerically (section 2.4.1).

Having characterized the stochastic process exhibited by the tetrahedronmodel as semi-Markovian, a faster and more general applicable Monte Carlosimulation algorithm than the one used in ref. [103] was devised in section2.4.2. It is an extension of the famous Gillespie algorithm ([37]) and thereforedoes not rely on a time discretization which makes it both exact and costeffective. There have been, except for this work, no applications of semi-Markovprocesses to cell biology so far, at least to the authors knowledge. However,they are known for a long time and in the context of machine learning andhidden semi-Markov processes have been successfully applied to various areas,such as speech recognition or human activity prediction [115]. As the HSMmodeling scheme naturally leads to a semi-Markovian description, there might befurther applications to stochastic cellular processes where a state space reductionis inevitable for feasible modeling. For example, to describe stochastic geneexpression a standard Markovian description bears the risk of oversimplificationwhich could be attenuated by a semi-Markovian approach.

In the last section 2.5 the predictive power of the presented Ca2+ signalingmodeling approach could be demonstrated. Calculations of the cellular responsecharacterized by the average ISIs with respect to two different stimuli, predicteda fold change encoding of stimulus intensity changes. This paradigm explainsvery consistent how a cell population with large cell-to-cell variability in spikingpatterns coherently reacts to a changing extracellular signal. Strikingly, thisbehavior could be recently established experimentally in close collaboration withthe Taylor lab [105]. The experimental data allowed for a more phenomenologicaldescription of the cellular response towards extracellular stimuli. The deducedexponential relationship between Tav and the stimulus concentration proved alsoto be applicable to different cell and stimulus types. It relies on the independenceof the fold change value β from the reference stimulus intensity. This is notcaptured by the Ca2+ modeling so far, and closing this gap demands further

94

2.6. Discussion of the stochastic Ca2+ modeling

theoretical efforts.As Ca2+ serves as a versatile messenger molecule, the bigger question of course

is how the adaptive stochastic spike sequences get decoded to ultimately changethe cellular state, e.g. altering the expression profile or changing the functionalstate of enzymes, in a coordinated manner. There have been numerous studieson the downstream effects of intracellular calcium [9], a prominent exampleconcerns the transcription factor NFAT which is important for an effectiveimmune response [48]. It gets activated by the phosphatase calcineurin whichactivity in turn is calcium dependent. As calcium signals were predominantlyviewed as oscillatory, many decoding models involve frequency decoding [69, 84],which is naturally unsuitable for stochastic signals. Other decoding modelsrely on signal integrators [18], but these are inconsistent with the large cell-to-cell variability of Tav . In summary, elaborating a coherent decoding modelincorporating the stochastic nature of the Ca2+ signals as well as the fold changeencoding of stimulus intensities is a challenging but essential task for the future.

95

3. Concluding remarks

Every model, irrespective of its elaborateness, should prove itself by reasonablepredictions. Subsequent experiments based on theoretical predictions which suc-cessfully establish new findings are challenging and rare, especially in molecularbiology. Nevertheless, this often proved to be a very effective and insightfulway of doing science. With respect to the studies presented here, it could beachieved in the field of Ca2+ signaling in a concerted effort by the Falcke groupand the experimentalists from the Taylor lab [105]. For p53, correspondingnew successful experiments are still lacking. Although, the theoretical analy-sis conducted in this work provides advice for the focus of new experiments.Despite all differences between the p53 and the Ca2+ signaling systems, theversatility of the possible downstream effects of both pathways is astonishing.If there is an evolutionary design principle of cellular signaling systems, tosteadily augment the inputs, the information content and the scope of an alreadyestablished signaling pathway, is open to speculation. Subsequent theoreticaland experimental studies should and will reveal more details on cellular signalprocessing. Furthermore, positive feedbacks play a central role in both systems.As these generally have the potential to decouple signal strength from stimulusstrength, it appears plausible to expect that they are a common design principleof cellular signaling pathways. Finally, elaborate single cell experiments formedthe basis for all investigations presented here. They opened the door for a muchdeeper understanding of intracellular processes, and will continue to challengeand enlighten theoretical and experimental cell biologists in the future.

97

A. Appendix p53

A.1. Peak detection with wavelets

Most real life data is noisy, which generally limits the efficiency of featureextraction from the data. Especially peak detection is difficult when the noiseamplitude is high, as the number of missed peaks (false negatives) and erro-neously detected peaks (false positives) can easily spoil the results. A standardstrategy to deal with noisy data is to apply some form of smoothing. Thisnaturally will either reduce false negative detections (under-smoothing) or falsepositive detections (over-smoothing). It is generally impossible to achieve bothdesirable properties at the same time with a standard smoothing approach, whichtypically works by convoluting the data with some window (or sometimes calledfilter) function. The wavelet transform, however, has the favorable property ofsmoothing the data on many scales at once. This, as to be seen in the following,allows for a peak detection algorithm which is capable of optimizing both thefalse negative and the false positive detection rate.

Wavelets were originally developed for signal processing, to allow for a goodresolution in both time and frequency domain. They are extensively used forthe spectral analysis of times series in various fields, they are especially popularin the geosciences [50]. The continuous wavelet transform is formally defined as:

WΦ(t, s)[f ] =1√s

∫ +∞

−∞Φ∗( t − τ

s

)f(t)dτ (A.1)

Here f(t) is the signal and the Φs,τ (t) are the wavelets. The resulting waveletspectrum WΦ(t, s) is a two-dimensional time-frequency representation of theoriginal signal, where s denotes the scale and t the time. Wavelets are a familyof functions which can be used to construct a basis for the Hilbert space L2(R).An important property of wavelets is, that they have “finite energy” or as acorollary are square-integrable. Loosely speaking, this property stems from theirtime-localization (see example in figure A.1). This is for example not true for thebasis functions of the Fourier transform. Another important property concernsthe vanishing mean

∫+∞−∞ Φs,τ (t)dt = 0. A specific wavelet basis function can be

derived from a mother wavelet by two operations, dilation (s) and translation(τ), so that one may write Φs,τ (t) = 1√

sΦ(

t−τs

). An example for such a mother

wavelet is the Ricker wavelet which reads

Φ(t) =2√3π

14

(1 − t2

)e− t2

2 . (A.2)

It is the negative second derivative of the Gaussian function. By substitutingt → t−τ

s the whole family Φs,τ of Ricker wavelets can be derived from that

99

A. Appendix p53

equation.

Figure A.1.: The Ricker wavelet for τ = 0 and the scale parameter s as indicatedin the legend, the mother wavelet corresponds to s = 1.

To illustrate the peak detection algorithm, a synthetic trajectory is constructedout of a slow sinusoidal signal plus additive white Gaussian noise. Then, twoGaussian peaks are added, one narrow on top of a sinusoidal arc and one widerat the bottom. The test trajectory is shown below in figure A.2.

Visualizations of wavelet transforms are called scaleograms and are easy tocomprehend by noting that for a fixed scale s, equation A.1 constitutes theconvolution (i.e. an unnormalized cross-correlation) of the signal f(t) withthe wavelet function of scale s. Therefore, the scaleogram can be interpretedas stacked convolutions with increasing scale over some range from smin tosmax. A value of the spectrum at a point (t, s) corresponds to the unnormalizedcorrelation of the signal with the wavelet of scale s centered at t. Two horizontalslices, meaning two selected convolutions, of the wavelet transform are shown infigure A.3b.

The scale range [smin, smax] is very critical for the scaleogram. Wavelets withthe smallest scale determine the widths of the narrowest peaks which can still bedetected. Wavelets with larger scales are very good noise filters, but if too bigstart to capture global trends of the trajectory. Hence, wavelets have a bandpasscharacter. To illustrate this, two scaleograms of the synthetic trajectory withill-chosen scale ranges are shown in figure A.4.

To progress with the peak detection so called ridges are identified in thescaleogram. A ridge is the vertical connection of relative maxima within thestacked convolutions. This strategy was originally devised in the context offeature detection in mass spectra [24]. An important parameter for the ridgeline construction controls the maximal acceptable horizontal distance (i.e. the

100


Figure A.2.: The synthetic test trajectory. White noise was added to a slowsinusoidal background signal. Then, the two peaks to detect areplaced at times 5 and 10 respectively. They are scaled Gaussianswith standard deviation σ = 0.1 and σ = 0.5. Its wavelet transformis shown in figure A.3a.

time-lag) of two relative extrema coming from adjacent convolutions to getconnected. Unsymmetric peaks, for example, will generally show a tilted ridgeline in the scaleogram, corresponding to a drift in time of the relative extrema forincreasing scales. Loosely speaking, a ridge line is a vertical, but not necessarilystraight, slice through the scaleogram. To give a real life example, the outputof the peak detection algorithm devised for the p53 data analysis is shown infigure A.5.

The final step of the peak detection is to filter the constructed ridge lines. Thisis an intricate process, as there are no a priori perfect filter criteria available. Apowerful criteria is obviously the maximum value of a ridge line, as it is effectivelyan amplitude filter. Also the length of a ridge line, as it is an indicator of thevertical extent of a feature in the scaleogram, can be very informative. Randomfluctuations usually have, if any, very short ridge lines. Howver, the performanceof the filter criteria are not independent on the scale range [smin, smax] andobviously not on the ridge line construction . So for an application to real data,many sensible adjustments depending on, among others, peak shapes, noise leveland sampling frequency have to be made.

The ridge line representation of a peak in the scaleogram is the core ideabehind the peak detection method presented here. It allows to assess a featuressignificance over many (smoothing) scales at once, and therefore allows a morerobust detection performance for highly irregular data. In conclusion, peakdetection with wavelets is incredibly versatile, but on the downside also moredifficult to master.

101

A. Appendix p53

(a) (b)

Figure A.3.: (a) The wavelet transform, or scaleogram, of the test trajectoryshown in figure A.2. The scale range used for the Ricker waveletsgoes from smin = 0.1 to smax = 0.6. Positive correlation is indicatedin red, negative correlation in blue. The two noticeable stripesof positive correlation are the wavelet representation of the peaksto detect in the trajectory. (b) Two horizontal slices through thescaleogram in (a). These correspond to two convolutions of thesignal with the Ricker wavelet for two different scales as indicated inthe legend. Wavelets with larger scales naturally smooth the datamore, but have problems to distinguish the sinusoidal backgroundfrom the narrow peaks. Smaller scales capture also narrow features,but are more prone to noise.

102


(a) (b)

Figure A.4.: Wavelet transforms of the synthetic trajectory shown in figureA.2 with ill-chosen scale ranges [smin, smax] (a) Scaleogram withthe scale range [0.05,0.2], the noise is not effectively filtered out.(b) Scaleogram with the scale range [0.2,3.], the noise is almostcompletely suppressed but the global sinusoidal background signaldominates the scaleogram.

103

A. Appendix p53

Figure A.5.: Example output of the peak detection program devised for the p53data analysis. The first plot shows the padded trajectory with thecolor coded detected peaks, it is from a cell stimulated with 100ngNCS (see main text). In the middle the corresponding wavelettransform (scaleogram) together with the successfully constructedridge lines is shown. The wavelet scale is given in relative unitswith smin ≡ 0 and smax ≡ 100. The last plot at the bottom depictsall identified ridge lines, which color correspond to the color codeddetected peaks in the trajectory. Ridge lines in black were filteredout. Optimally, a ridge line is a concave function of the waveletscale, with a maximum corresponding to the highest correlation ata scale associated with the respective peak width. The employedfilter criteria for the ridge lines are a minimum correlation of 0.2and a minimum length of 100, which corresponds to the total scalerange. The absolute scale range was smin = 2h and smax = 7h.

104

A.2. Table of Parameters for the p53 model

A.2. Table of Parameters for the p53 modelAll parameters used for the excitable p53 model and their values used for thesimulations.

Table A.1.: Overview of all model parameters

namemeaning value

A maximal self activation rate of ATM 30.5P rate of dephosphorylation of ATM by Wip1 22S rate of ATM∗ formation induced by DSB loci, damage signal

strength0 < S < 0.4

γ sensitivity parameter for the logarithmic transformS(DSB) = γ ln(DSB + 1)

0.06

C production rate of p53 1.4g maximal degradation of P53 by Mdm2 2.5dAM degradation rate of Mdm2 by ATM∗ 1.Tm maximal production rate of mdm2 1.TM maximal production rate of Mdm2 4.Tw maximal production rate of wip1 1.TW maximal production rate of Wip1 1.dA basal dephosphorylation rate of ATM∗ 0.16dP basal degradation rate of P53 0.1dm basal degradation rate of mdm2 1.dM basal degradation rate of Mdm2 2.dw basal degradation rate of wip1 1.3.dW basal degradation rate of Wip1 2.3kA Michaelis constant for the ATM∗ self activation 0.5kW A Michaelis constant for the inhibition of the ATM∗ self acti-

vation by Wip10.14

kMP Michaelis constant for the degradation of P53 by Mdm2 0.15kP m Michaelis constant for the production of mdm2 by P53 1.kP w Michaelis constant for the production of wip1 by P53 1.R strength of the inhibition of the Mdm2 mediated degradation

of P53 by ATM∗2

105

A. Appendix p53

For better readability the model equations again read:

d

dtATM∗ = A

ATM∗2

kA + ATM∗21

1 + Wip1/kW A− dAATM∗ − P ATM∗ Wip1 + S(DSB)

d

dtP53 = C − dP P53 − g Mdm2

P53kMP + P53

(1 +

R

1 + ATM∗

)d

dtmdm2 = Tm

P53kP m + P53

− dm mdm2

d

dtMdm2 = TM mdm2 − dM Mdm2 − dAM ATM∗ Mdm2

d

dtwip1 = Tw

P53kP w + P53

− dw wip1

d

dtWip1 = TW wip1 − dW Wip1

106

A.3. Sensitivity of pulse shapes and the excitation threshold on parameter variations

A.3. Sensitivity of pulse shapes and the excitationthreshold on parameter variations

Here numerical studies about the influence of the model parameters on theamplitude and the length of the pulses shall be presented. Additionally theexcitation threshold shall be quantified, and its parameter dependence analysed.In general the parameter plots shown in the following are qualitatively the samefor parameters of the same bifurcation type as introduced in section 1.5.2 of themain text and shown as table below. So there are again four plots sufficient toqualitatively characterize the parameter dependencies of the model.

Table A.2.: Overview of the bifurcation diagram type membership for all modelparameters

type I type II type III type IV

Parameters C, dm, dM

dAM , kmp

g, Tm, TM

RP, Tw, TW A, dw, dW

kW A, kP w

Trait positive onP53

negative onP53

negative onATM∗

positive onATM∗

For performing the pulse shape analysis, the system is initialized at the loweststable steady state plus a little initial kick in positive ATM∗ direction to crossthe excitation threshold. The parameter regions scanned always reach fromhalf up to twice the respective default value. It is noteworthy, that the regionof excitability is bigger than the region characterized by one stable and twounstable fixed points. The phasespace still feels the existence of the unstablefixed points which disappeared via a saddle-node bifurcation. Therefore, theseplots also directly indicate the total parameter region of excitability. Thisadditional region of excitable behavior in parameter space is marked by blackdots in the following plots.

The results for the amplitudes and pulse lengths shown in figures A.6 and A.7indicate that different parameters show a varying degree of influence on the p53pulse amplitudes. Although the qualitative pulse shape dependencies are thesame for all parameters belonging to the same type, the effects may significantlydiffer in quantity. By analysing all parameters like described and shown above,their impact on the pulse shapes was classified as either strong or weak. TableA.3 gives a general overview of the sensitivity for the main model parameters.

To test the model parameter influence on the excitation threshold, all speciesare initialized at the stable steady state, except for ATM which is taken outof an interval [0, ATM∗

max]. For each set of the initial conditions the resultingtrajectory is tested for an excitation loop. This procedure is repeated to scanthrough the same range of parameter values as before and the minimal necessaryATM∗ value for triggering a pulse is recored respectively. The results are shown

107

A. Appendix p53



Figure A.6.: Model parameter influence on p53 pulse amplitudes. The showndependencies are qualitatively the same for parameters belongingto the same type according to table A.2.

in figure A.8, black dotes again depict the enlarged region of excitability withoutthree fixes points.

The slope of the threshold dependence can generally be inferred from thecodimension-2 bifurcation sets shown in the subsequent section.

108




Figure A.7.: Model parameter influence on p53 pulse length. The shown depen-dencies are qualitatively the same for parameters belonging to thesame type according to table A.2.

109

A. Appendix p53

Table A.3.: Overview of the impact of parameter variations with respect to p53pulse shapes for the main model parameters

parameter amplitude length

A strong strongP strong strongC strong weakg weak weakdAM weak weakTm weak weakTM weak weakTw strong strongTW strong strongdM weak weakdW strong strongdP strong weakR weak weakS weak weak

110




Figure A.8.: Selected parameter influence on the excitation threshold. Theshown dependencies are qualitatively the same for parametersbelonging to the same type according to table A.2.

111

A. Appendix p53

A.4. Codimension-2 bifurcation diagramsAn effective way to depict the model regimes and parameter dependencies arebifurcation set which trace the saddle-node curves and the Hopf-curve. Theexcitable regime always spreads from the Cusp point, denoted by ’CP1’ in theplots, in between the two saddle-node curves. It is additionally confined by theHopf-curve which crosses one of the two saddle-node curves, see also figure 1.31in the main text.

(a) C, type I (b) g, type II

(c) TW , type III (d) dW , type IV

Figure A.9.: Bifurcation sets for different p53 model parameters. The regionof excitability always spreads from the Cusp point in between thetwo saddle-node curves, see also figure 1.31 in the main text. Forbetter visibility of the excitable region a logarithmic scale for thesignal strength S was chosen here.

112

A.5. Period of oscillations in the p53 model

A.5. Period of oscillations in the p53 modelAs outlined in the main text, the period of the limit cycle appearing in the p53model tends to infinity at the homoclinic bifurcation point. However, in practicethe period dependence of the limit cycle on the bifurcation parameter is verysteep. As shown in figure A.10, the period changes only very little over most ofthe parameter range.

Figure A.10.: Period of the limit cycle oscillations in dependence on the bifurca-tion parameter S. Over most of the parameter range, the periodchanges very little. Only very close to the homoclinic bifurcationpoint, it grows rapidly and tends to infinity.

113

B. Appendix Ca2+

B.1. Laplace transformations of Ψo and Ψc

The generalized exponential (GE) distribution is a generalized form of the humanmortality distribution discovered by Gompertz and Verhulst in the first half ofthe 19th century [39, 42]. For the analytic solution of the first passage timeproblem formulated in the main text, the Laplace transformation of the GEdensity function and its survival function are needed. They constitute the non-exponential waiting times used for the opening transitions of the tetrahedronmodel.

The GE density itself reads

Ψo(t) = αλ (1 − e−λt)α−1 e−λt. (B.1)

The Laplace transformation of a function f(t) is defined by

L{f}(s) =∫ ∞

0f(t) e−stdt, (B.2)

and so the following integral needs to be solved:

L{Ψo}(s) =∫ ∞

0αλ (1 − e−λt)α−1 e−λt e−stdt. (B.3)

Substituting y = e−t gives

αλ

∫ 1

0(1 − yλ)α−1 yλ+s−1dy. (B.4)

Next the substitution x = yλ is applied:

α

∫ 1

0(1 − x)α−1 x

sλ dx. (B.5)

Now the definition of the beta function is used, a special function alreadystudied by Euler. Written as an integral, it reads:

B(k, l) =∫ 1

0(1 − x)k−1 xl−1dx. (B.6)

There is also a form involving the Euler Γ-function:

B(k, l) =Γ(k)Γ(l)Γ(k + l)

. (B.7)

115

B. Appendix Ca2+

By using l = sλ + 1 and substituting k = α, one obtains from Eq. B.5:

L{Ψo}(s) =αΓ(α)Γ( s

λ + 1)Γ(α + s

λ + 1). (B.8)

The Laplace transformation for the survival function

Ψo(t) = 1 − (1 − e−λt)α (B.9)

can be done analogically. This time the following integral needs to be solved:

L{Ψo}(s) =1s

−∫ ∞

0(1 − e−λt)αdt, (B.10)

where the trivial Laplace transformation of a constant was already used. Substi-tuting y = e−t followed by the second substitution x = yλ yields

1s

− 1λ

∫ 1

0(1 − x)α x

sλ

−1dx. (B.11)

Now the the beta function (Eq. B.7) is recalled and a final substitution l = sλ

gives the solution:

L{Ψo}(s) =1s

− Γ(α + 1)Γ( sλ)

λΓ(α + sλ + 1)

, (B.12)

where k = α + 1.

The powers of Ψo needed for the construction of the conditioned waiting timescan be written as

Ψno =

n∑k=0

(n

k

)(−1)k(1 − e−λt)kα. (B.13)

These are just sums of the original terms with a rescaled shape parameter kα,and therefore their Laplace transforms are sums of the accordingly rescaledresults given above in Eq. B.12.

For the purposes of hierarchic stochastic modelling, the Laplace transform ofthe closing waiting time is also required. By using the binomial theorem, onemay rewrite Ψc as:

Ψc(t) = Nchγe−γtNch−1∑

k=0

(Nch − 1

k

)(−1)ke−kγt, (B.14)

where 1Nch−1−k = 1 was dropped inside the sum. After multiplication one gets:

Ψc(t) = NchγNch−1∑

k=0

(Nch − 1

k

)(−1)ke−(k+1)γt. (B.15)

In the end just the Laplace transform of a sum of exponentials is needed, and

116

B.1. Laplace transformations of Ψo and Ψc

one finally obtains:

Ψc(s) = NchγNch−1∑

k=0

(Nch − 1

k

)(−1)k (−1)k

s + (k + 1)γ. (B.16)

This form of the Laplace transformed closing time can be used for constructionof the conditioned waiting times, which are needed for computation of the Ca2+

spiking statistics by Eq. 2.34 of the main text.

117

C. List of abbreviations

FP fixed pointr.h.s. right hand sideATM ataxia telangiectasia mutatedDSB DNA double strand breakDDR DNA damage responseIPI inter pulse intervalODE ordinary differential equationNCS neocarzinostatinLCO limit cycle oscillatorTF transcription factorl.h.s. left hand sideWm Wortmannin

ISI inter spike intervalTav average ISIHSM hierarchic stochastic modellingGE generalized exponentialCV coefficient of variationHEK human embryonic kidney 293 cellsCCh carbachol

119

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129

Acknowledgments

Many people contributed, on one way or the other, to the process of creatingthis scholarly piece of work.

Dr. Martin Falcke is a supervisor, who is not afraid to engross his thoughts inthe very details of the students considerations. His rigorous analytic thoughtsand analysis often served as a very helpful paradigm for me, and especially inthe beginning guided me on how approach biology with the methodology andmindset of a theoretical physicist. Additionally, he also provided many thoughtsabout the bigger picture of Ca2+ signaling or science in general. I am deeplythankful for his trust and his sometimes sudden bursts of valuable ideas.

Professor Hanspeter Herzel provided with his long and rich experience thenarratives every scientist needs, to orientate and locate himself and his workwithin the scientific community. His farsightedness and universal knowledge oftenprovided very helpful linkages to other studies and insights from various fields,and steered me away from moments of tunnel vision every student experiences atone time or the other. He particularly raised my awareness for the importanceof data and data analysis, and the subtle pitfalls and ambiguities a theoreticalbiologist needs to know about. I am very thankful for his guiding spirit, whichhelped me to master the challenges of this thesis.

Without the participation of Dr. Alexander Löwer the project about p53would simply not have happened. He gave me the opportunity to directlycollaborate with experimentalists, which value can not be overstated. I amespecially thankful for the many fruitful and intellectual challenging discussionswhich introduced me to the mindset and language of experimental biologists,and allowed me to develop thoughts and questions a theoretician alone wouldnever think of. Even though he often grounded my overly ambitious plans, hemade me realize that real progress can be made primarily in collaboration withthe people who actually interact with the cells in the laboratory.

I would also like to thank the CSB graduate school for providing the fundingand the many possibilities of exchange of views with other PhD students andPIs.

I am much obliged to the open source community around the python program-ming language, especially the SciPy and PyDStool community, for providing theindispensable tools needed for this work. The realisation of a helpful, and yesoften anonymous but also altruistic and fun, community often lifted my spiritand motivated me to contribute in the future.

I would also like to thank all my Berlin dwellers who enriched my life experienceprofoundly. I want to single out my dear friend Matthieu Séry, who as an artistof life and art provided invaluable guidance on how to preserve and extend onesown identity during the progress of life.

Ich möchte mich auch bei meiner Mutter bedanken, war sie es doch die schon

131

Acknowledgments

in recht jungen Jahren meine Begeisterung für die große und interessante Weltder Wissenschaft im allgemeinen, und der Molekularbiologie im besonderen,geweckt hat. Auch meine Versuche sich als Physiker im Wirrwarr der Zellbiologiezurechtzufinden, hat sie bis zuletzt immer unterstützt.

Nun kann ich endlich auch meiner lieben Theresa danken. Ihr lebensfrohesWesen, ihre Zuneigung und ihr eben nicht nur emotionales Verständnis warenmir unverzichtbare Unterstützung in den vergangenen Jahren. Keiner hat dieFreuden und Leiden dieser Promotion intensiver begleitet als sie. Ich hoffe sehr,dass bald ich ihr auf diesem einzigartigen Abschnitt des Lebensweges ebensoden Rücken und das Herz stärken kann.

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stochastic oscillations in living cells

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